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solution.tex
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%
\documentclass{book}
\usepackage[utf8]{inputenc}
\usepackage{geometry}
\usepackage{amsmath}
\usepackage{tikz}
\usepackage{gensymb}
\usepackage{physics}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{hyperref}
\usepackage{extarrows}
\usepackage{dsfont}
\usepackage{cancel}
\usepackage{graphicx}
\usetikzlibrary{calc}
\geometry{a4paper}
\linespread{1.25}
\allowdisplaybreaks
\renewcommand\labelenumi{(\alph{enumi})}
\renewcommand\labelenumii{(\roman{enumii})}
\title{Solution to Topological Quantum: Lecture Notes and Proto-Book}
\author{Jiaqi Guo\footnote{Email: \href{mailto:[email protected]}{[email protected]}.}}
%\date{December 2022}
\begin{document}
\maketitle
\tableofcontents
\chapter{Introduction: History of Topology, Knots, Peter Tait and Lord Kelvin}
\chapter{Kauffman Bracket Invariant and Relation to Physics}
\section{Trefoil Knot and the Kauffman Bracket}
Using the Kauffman rules, calculate the Kauffman bracket invariant of the right and left handed trefoil knots shown in Fig. \ref{fig:leftAndRightHandedTrefoil}. Conclude these two knots are topologically inequivalent. While this statement appears obvious on sight, it was not proved mathematically until 1914 (by Max Dehn). It is trivial using this technique!
\begin{figure}[h!]
\centering
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\end{tikzpicture}
\caption{Left and Right Handed Trefoil Knots (on the left and right respectively)}
\label{fig:leftAndRightHandedTrefoil}
\end{figure}
\paragraph{Remark} The word “trefoil” is from the plant trifolium, or clover, which has compound trifoliate leaves.
\paragraph{Answer}
We try to figure out the right trefoil first:
\begin{align*}
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\\
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\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (23,30.32) .. controls (28.66,23.32) and (30.66,14.66) .. (36,18.99) .. controls (41.33,23.32) and (40,33.82) .. (33.5,34.99) .. controls (26.99,36.16) and (22.83,33.32) .. (18.5,30.32) .. controls (14.16,27.32) and (11.67,15.29) .. (5.66,18.99) .. controls (-0.35,22.7) and (2.83,39.99) .. (16.5,34.32) ;
%Curve Lines [id:da372717649034797]
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (20.16,14.49) .. controls (34.5,14.16) and (27.66,1.99) .. (20.16,1.99) .. controls (12.66,1.99) and (6.33,14.99) .. (20.16,14.49) -- cycle ;
\end{tikzpicture}
\right)\\
= & A^{2}\left( A\tikzset{every picture/.style={line width=0.75pt}} %set default line width to 0.75pt
\begin{tikzpicture}[x=0.75pt,y=0.75pt,yscale=-1,xscale=1, baseline=(XXXX.south) ]
\path (0,40);\path (43.99652099609375,0);\draw ($(current bounding box.center)+(0,0.3em)$) node [anchor=south] (XXXX) {};
%Shape: Circle [id:dp6253531078371917]
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (5.3,20.64) .. controls (5.3,11.06) and (13.06,3.29) .. (22.65,3.29) .. controls (32.23,3.29) and (40,11.06) .. (40,20.64) .. controls (40,30.22) and (32.23,37.99) .. (22.65,37.99) .. controls (13.06,37.99) and (5.3,30.22) .. (5.3,20.64) -- cycle ;
%Shape: Circle [id:dp34206862302889895]
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (14.97,20.64) .. controls (14.97,16.4) and (18.41,12.96) .. (22.65,12.96) .. controls (26.88,12.96) and (30.32,16.4) .. (30.32,20.64) .. controls (30.32,24.88) and (26.88,28.32) .. (22.65,28.32) .. controls (18.41,28.32) and (14.97,24.88) .. (14.97,20.64) -- cycle ;
\end{tikzpicture}
+A^{-1}\tikzset{every picture/.style={line width=0.75pt}} %set default line width to 0.75pt
\begin{tikzpicture}[x=0.75pt,y=0.75pt,yscale=-1,xscale=1, baseline=(XXXX.south) ]
\path (0,40);\path (43.99652099609375,0);\draw ($(current bounding box.center)+(0,0.3em)$) node [anchor=south] (XXXX) {};
%Shape: Circle [id:dp3384239181781197]
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (5.3,21.64) .. controls (5.3,12.06) and (13.06,4.29) .. (22.65,4.29) .. controls (32.23,4.29) and (40,12.06) .. (40,21.64) .. controls (40,31.22) and (32.23,38.99) .. (22.65,38.99) .. controls (13.06,38.99) and (5.3,31.22) .. (5.3,21.64) -- cycle ;
\end{tikzpicture}
\right) +\left( A\tikzset{every picture/.style={line width=0.75pt}} %set default line width to 0.75pt
\begin{tikzpicture}[x=0.75pt,y=0.75pt,yscale=-1,xscale=1, baseline=(XXXX.south) ]
\path (0,40);\path (43.99652099609375,0);\draw ($(current bounding box.center)+(0,0.3em)$) node [anchor=south] (XXXX) {};
%Shape: Circle [id:dp5240073107156691]
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (5.3,21.64) .. controls (5.3,12.06) and (13.06,4.29) .. (22.65,4.29) .. controls (32.23,4.29) and (40,12.06) .. (40,21.64) .. controls (40,31.22) and (32.23,38.99) .. (22.65,38.99) .. controls (13.06,38.99) and (5.3,31.22) .. (5.3,21.64) -- cycle ;
\end{tikzpicture}
+A^{-1}\tikzset{every picture/.style={line width=0.75pt}} %set default line width to 0.75pt
\begin{tikzpicture}[x=0.75pt,y=0.75pt,yscale=-1,xscale=1, baseline=(XXXX.south) ]
\path (0,31);\path (59.99232482910156,0);\draw ($(current bounding box.center)+(0,0.3em)$) node [anchor=south] (XXXX) {};
%Shape: Circle [id:dp17633990879069805]
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (3.3,16.14) .. controls (3.3,10.15) and (8.15,5.29) .. (14.15,5.29) .. controls (20.14,5.29) and (25,10.15) .. (25,16.14) .. controls (25,22.13) and (20.14,26.99) .. (14.15,26.99) .. controls (8.15,26.99) and (3.3,22.13) .. (3.3,16.14) -- cycle ;
%Shape: Circle [id:dp09348561797596755]
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (32.3,16.14) .. controls (32.3,10.15) and (37.15,5.29) .. (43.15,5.29) .. controls (49.14,5.29) and (54,10.15) .. (54,16.14) .. controls (54,22.13) and (49.14,26.99) .. (43.15,26.99) .. controls (37.15,26.99) and (32.3,22.13) .. (32.3,16.14) -- cycle ;
\end{tikzpicture}
\right)\\
& +\left( A\tikzset{every picture/.style={line width=0.75pt}} %set default line width to 0.75pt
\begin{tikzpicture}[x=0.75pt,y=0.75pt,yscale=-1,xscale=1, baseline=(XXXX.south) ]
\path (0,40);\path (43.99652099609375,0);\draw ($(current bounding box.center)+(0,0.3em)$) node [anchor=south] (XXXX) {};
%Shape: Circle [id:dp7807071920749482]
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (5.3,20.64) .. controls (5.3,11.06) and (13.06,3.29) .. (22.65,3.29) .. controls (32.23,3.29) and (40,11.06) .. (40,20.64) .. controls (40,30.22) and (32.23,37.99) .. (22.65,37.99) .. controls (13.06,37.99) and (5.3,30.22) .. (5.3,20.64) -- cycle ;
%Shape: Circle [id:dp20643797150504994]
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (14.97,20.64) .. controls (14.97,16.4) and (18.41,12.96) .. (22.65,12.96) .. controls (26.88,12.96) and (30.32,16.4) .. (30.32,20.64) .. controls (30.32,24.88) and (26.88,28.32) .. (22.65,28.32) .. controls (18.41,28.32) and (14.97,24.88) .. (14.97,20.64) -- cycle ;
\end{tikzpicture}
+A^{-1}\tikzset{every picture/.style={line width=0.75pt}} %set default line width to 0.75pt
\begin{tikzpicture}[x=0.75pt,y=0.75pt,yscale=-1,xscale=1, baseline=(XXXX.south) ]
\path (0,31);\path (59.99232482910156,0);\draw ($(current bounding box.center)+(0,0.3em)$) node [anchor=south] (XXXX) {};
%Shape: Circle [id:dp14222723555272965]
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (3.3,16.14) .. controls (3.3,10.15) and (8.15,5.29) .. (14.15,5.29) .. controls (20.14,5.29) and (25,10.15) .. (25,16.14) .. controls (25,22.13) and (20.14,26.99) .. (14.15,26.99) .. controls (8.15,26.99) and (3.3,22.13) .. (3.3,16.14) -- cycle ;
%Shape: Circle [id:dp13034301114351332]
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (32.3,16.14) .. controls (32.3,10.15) and (37.15,5.29) .. (43.15,5.29) .. controls (49.14,5.29) and (54,10.15) .. (54,16.14) .. controls (54,22.13) and (49.14,26.99) .. (43.15,26.99) .. controls (37.15,26.99) and (32.3,22.13) .. (32.3,16.14) -- cycle ;
\end{tikzpicture}
\right) +A^{-2}\left( A\tikzset{every picture/.style={line width=0.75pt}} %set default line width to 0.75pt
\begin{tikzpicture}[x=0.75pt,y=0.75pt,yscale=-1,xscale=1, baseline=(XXXX.south) ]
\path (0,44);\path (26.994884490966797,0);\draw ($(current bounding box.center)+(0,0.3em)$) node [anchor=south] (XXXX) {};
%Shape: Circle [id:dp8462936980147189]
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (12.93,2.79) .. controls (17.43,2.79) and (21.07,6.43) .. (21.07,10.93) .. controls (21.07,15.42) and (17.43,19.06) .. (12.93,19.06) .. controls (8.44,19.06) and (4.8,15.42) .. (4.8,10.93) .. controls (4.8,6.43) and (8.44,2.79) .. (12.93,2.79) -- cycle ;
%Shape: Circle [id:dp8544095207014937]
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (12.93,24.54) .. controls (17.43,24.54) and (21.07,28.18) .. (21.07,32.67) .. controls (21.07,37.17) and (17.43,40.81) .. (12.93,40.81) .. controls (8.44,40.81) and (4.8,37.17) .. (4.8,32.67) .. controls (4.8,28.18) and (8.44,24.54) .. (12.93,24.54) -- cycle ;
\end{tikzpicture}
+A^{-1}\tikzset{every picture/.style={line width=0.75pt}} %set default line width to 0.75pt
\begin{tikzpicture}[x=0.75pt,y=0.75pt,yscale=-1,xscale=1, baseline=(XXXX.south) ]
\path (0,42);\path (47.98960876464844,0);\draw ($(current bounding box.center)+(0,0.3em)$) node [anchor=south] (XXXX) {};
%Shape: Circle [id:dp2808548197181753]
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (22.93,1.96) .. controls (27.43,1.96) and (31.07,5.61) .. (31.07,10.1) .. controls (31.07,14.6) and (27.43,18.24) .. (22.93,18.24) .. controls (18.44,18.24) and (14.8,14.6) .. (14.8,10.1) .. controls (14.8,5.61) and (18.44,1.96) .. (22.93,1.96) -- cycle ;
%Shape: Circle [id:dp4701752603617959]
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (10.93,23.71) .. controls (15.43,23.71) and (19.07,27.36) .. (19.07,31.85) .. controls (19.07,36.34) and (15.43,39.99) .. (10.93,39.99) .. controls (6.44,39.99) and (2.8,36.34) .. (2.8,31.85) .. controls (2.8,27.36) and (6.44,23.71) .. (10.93,23.71) -- cycle ;
%Shape: Circle [id:dp8102265525674299]
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (35.86,23.71) .. controls (40.35,23.71) and (43.99,27.35) .. (43.99,31.85) .. controls (43.99,36.34) and (40.35,39.98) .. (35.86,39.98) .. controls (31.36,39.98) and (27.72,36.34) .. (27.72,31.85) .. controls (27.72,27.35) and (31.36,23.71) .. (35.86,23.71) -- cycle ;
\end{tikzpicture}
\right)\\
= & A^{2} (Ad^{2} +A^{-1} d)+(Ad+A^{-1} d^{2} )+(Ad +A^{-1} d^{2} )+A^{-2} (Ad^{2} +A^{-1} d^{3} )\\
= & -A^{-9} +A+A^{3} +A^{7} .
\end{align*}
Similar calculation gives
\begin{equation*}
\tikzset{every picture/.style={line width=0.75pt}} %set default line width to 0.75pt
\begin{tikzpicture}[x=0.75pt,y=0.75pt,yscale=-1,xscale=1, baseline=(XXXX.south) ]
\path (0,68);\path (72.99199676513672,0);\draw ($(current bounding box.center)+(0,0.3em)$) node [anchor=south] (XXXX) {};
%Curve Lines [id:da43968276064415623]
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (32.37,55.68) .. controls (22.67,42.88) and (8.2,19.54) .. (20.17,8.02) .. controls (32.14,-3.5) and (57.89,4.18) .. (54.5,21.57) ;
%Curve Lines [id:da5050406190521861]
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (52.98,30.15) .. controls (46.54,44.85) and (33.23,68.88) .. (17.33,64.05) .. controls (1.43,59.22) and (-4.43,33) .. (12.4,27.47) ;
%Curve Lines [id:da7698083169944674]
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (20.21,25.25) .. controls (36.1,22.96) and (63.53,21.6) .. (67.82,37.65) .. controls (72.1,53.7) and (52.92,72.51) .. (39.34,61.13) ;
\end{tikzpicture}
=-A^{9} +A^{-1} +A^{-3} +A^{-7} .
\end{equation*}
We can see the mirror symmetry gives $A\rightarrow A^{-1}$. In other notations, we can set $d^{i}\rightarrow d^{i-1}$, then the trivial knot(i.e. a circle) give $1$. Then we have
\begin{equation*}
\tikzset{every picture/.style={line width=0.75pt}} %set default line width to 0.75pt
\begin{tikzpicture}[x=0.75pt,y=0.75pt,yscale=-1,xscale=1, baseline=(XXXX.south) ]
\path (0,68);\path (72.99199676513672,0);\draw ($(current bounding box.center)+(0,0.3em)$) node [anchor=south] (XXXX) {};
%Curve Lines [id:da9633941749826875]
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (38.33,55.68) .. controls (48.02,42.88) and (62.5,19.54) .. (50.52,8.02) .. controls (38.55,-3.5) and (12.8,4.18) .. (16.19,21.57) ;
%Curve Lines [id:da6880787601193201]
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (17.72,30.15) .. controls (24.15,44.85) and (37.47,68.88) .. (53.37,64.05) .. controls (69.26,59.22) and (75.13,33) .. (58.29,27.47) ;
%Curve Lines [id:da8809202495162078]
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (50.49,25.25) .. controls (34.6,22.96) and (7.16,21.6) .. (2.88,37.65) .. controls (-1.41,53.7) and (17.78,72.51) .. (31.36,61.13) ;
\end{tikzpicture}
=A^{-7} -A^{5} -A^{3} .
\end{equation*}
\section{Abelian Kauffman Anyons}
Anyons described by the Kauffman bracket invariant with certain special values of the constant $A$ are abelian anyons - meaning that an exchange introduces only a simple phase as shown in Fig. \ref{fig:AbelianAnyonExchange}.
\begin{figure}[h!]
\centering
\tikzset{every picture/.style={line width=0.75pt}} %set default line width to 0.75pt
\begin{tikzpicture}[x=0.75pt,y=0.75pt,yscale=-1,xscale=1]
%uncomment if require: \path (0,68); %set diagram left start at 0, and has height of 68
%Straight Lines [id:da12351686453929012]
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (240.67,3.65) -- (303.37,66.35) ;
%Straight Lines [id:da5675480043864236]
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (278.45,27.79) -- (303.45,2.79) ;
%Straight Lines [id:da37770163193150785]
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (240.11,66.12) -- (265.11,41.12) ;
%Curve Lines [id:da4265407093388114]
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (369.17,3.82) .. controls (381.5,32.65) and (381.17,38.65) .. (369.17,65.32) ;
%Curve Lines [id:da5245986537990159]
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (415.96,3.82) .. controls (403.63,32.65) and (403.96,38.65) .. (415.96,65.32) ;
% Text Node
\draw (318.81,27.07) node [anchor=north west][inner sep=0.75pt] {$=\mathrm{e}^{\mathrm{i} \theta }$};
\end{tikzpicture}
\caption{For abelian anyons, exchange gives a phase $\mathrm{e}^{\mathrm{i} \theta }$.}
\label{fig:AbelianAnyonExchange}
\end{figure}
\begin{enumerate}
\item For $A=\pm \mathrm{e}^{\mathrm{i} \pi /3}$ (and the complex conjugates of these values), show that the anyons are bosons or fermions respectively (i.e., $\mathrm{e}^{\mathrm{i} \theta } =\pm 1$ ).
\item For $A=\pm \mathrm{e}^{\mathrm{i} \pi /6}$ (and the complex conjugates of these values) show the anyons are semions (i.e., $\mathrm{e}^{\mathrm{i} \theta } =\pm \mathrm{i}$ ). In fact these are precisely the anyons that arise for the $\nu =1/2$ fractional quantum Hall effect of bosons (We will discuss this later in this book (See chapter 37). This particular phase of quantum Hall matter has been produced experimentally(Clark et al. [2020]), but only in very small puddles so far and it has not been possible to measure braiding statistics as of yet.
\end{enumerate}
HINT: For (a) and (b) show first the identity shown in Fig. \ref{fig:ExchangeBosonsOrFermions}.
If you can't figure it out, try evaluating the Kauffman bracket invariant for a few knots with these values of $A$ and see how the result arises.
\begin{figure}[h!]
\centering
\tikzset{every picture/.style={line width=0.75pt}} %set default line width to 0.75pt
\begin{tikzpicture}[x=0.75pt,y=0.75pt,yscale=-1,xscale=1]
%uncomment if require: \path (0,65); %set diagram left start at 0, and has height of 65
%Curve Lines [id:da8961261533012252]
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (305.81,4.41) .. controls (275.96,19.3) and (269.75,18.9) .. (242.15,4.41) ;
%Curve Lines [id:da47083895189428304]
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (305.81,60.89) .. controls (275.96,46.01) and (269.75,46.41) .. (242.15,60.89) ;
%Curve Lines [id:da9898702962562469]
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (373.17,1.9) .. controls (385.5,30.74) and (385.17,36.74) .. (373.17,63.4) ;
%Curve Lines [id:da26291761978783246]
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (419.96,1.9) .. controls (407.63,30.74) and (407.96,36.74) .. (419.96,63.4) ;
% Text Node
\draw (324,25.65) node [anchor=north west][inner sep=0.75pt] {$=\pm $};
\end{tikzpicture}
\caption{For bosons or fermions the sign in this figure is $+$, for semions the sign is $-$.}
\label{fig:ExchangeBosonsOrFermions}
\end{figure}
\paragraph{Answer}
Using the definition of Kauffman bracket:
\begin{equation*}
\begin{aligned}
\tikzset{every picture/.style={line width=0.75pt}} %set default line width to 0.75pt
\begin{tikzpicture}[x=0.75pt,y=0.75pt,yscale=-1,xscale=1, baseline=(XXXX.south) ]
\path (0,68);\path (68.99652099609375,0);\draw ($(current bounding box.center)+(0,0.3em)$) node [anchor=south] (XXXX) {};
%Straight Lines [id:da5470670447844541]
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (3.67,4) -- (66.37,66.7) ;
%Straight Lines [id:da013140972627327496]
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (41.45,28.14) -- (66.45,3.14) ;
%Straight Lines [id:da8599231631706779]
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (3.11,66.47) -- (28.11,41.47) ;
\end{tikzpicture}
& =A\tikzset{every picture/.style={line width=0.75pt}} %set default line width to 0.75pt
\begin{tikzpicture}[x=0.75pt,y=0.75pt,yscale=-1,xscale=1, baseline=(XXXX.south) ]
\path (0,68);\path (53.991668701171875,0);\draw ($(current bounding box.center)+(0,0.3em)$) node [anchor=south] (XXXX) {};
%Curve Lines [id:da37581413358541704]
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (4.17,2.29) .. controls (16.5,31.12) and (16.17,37.12) .. (4.17,63.79) ;
%Curve Lines [id:da9006964162238551]
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (50.96,2.29) .. controls (38.63,31.12) and (38.96,37.12) .. (50.96,63.79) ;
\end{tikzpicture}
+A^{-1}\tikzset{every picture/.style={line width=0.75pt}} %set default line width to 0.75pt
\begin{tikzpicture}[x=0.75pt,y=0.75pt,yscale=-1,xscale=1, baseline=(XXXX.south) ]
\path (0,67);\path (71.98169708251953,0);\draw ($(current bounding box.center)+(0,0.3em)$) node [anchor=south] (XXXX) {};
%Curve Lines [id:da0002888383512205106]
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (68.81,4.31) .. controls (38.96,19.2) and (32.75,18.79) .. (5.15,4.31) ;
%Curve Lines [id:da2642472194401073]
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (68.81,60.79) .. controls (38.96,45.9) and (32.75,46.3) .. (5.15,60.79) ;
\end{tikzpicture}
\\
& \stackrel{!}{=}\mathrm{e}^{\mathrm{i} \theta }\tikzset{every picture/.style={line width=0.75pt}} %set default line width to 0.75pt
\begin{tikzpicture}[x=0.75pt,y=0.75pt,yscale=-1,xscale=1, baseline=(XXXX.south) ]
\path (0,68);\path (53.991668701171875,0);\draw ($(current bounding box.center)+(0,0.3em)$) node [anchor=south] (XXXX) {};
%Curve Lines [id:da5214125017288944]
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (4.17,2.29) .. controls (16.5,31.12) and (16.17,37.12) .. (4.17,63.79) ;
%Curve Lines [id:da2916432495183652]
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (50.96,2.29) .. controls (38.63,31.12) and (38.96,37.12) .. (50.96,63.79) ;
\end{tikzpicture}
,
\end{aligned}
\end{equation*}
which means we have:
\begin{equation*}
\tikzset{every picture/.style={line width=0.75pt}} %set default line width to 0.75pt
\begin{tikzpicture}[x=0.75pt,y=0.75pt,yscale=-1,xscale=1, baseline=(XXXX.south) ]
\path (0,68);\path (53.991668701171875,0);\draw ($(current bounding box.center)+(0,0.3em)$) node [anchor=south] (XXXX) {};
%Curve Lines [id:da3520499283776384]
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (4.17,2.29) .. controls (16.5,31.12) and (16.17,37.12) .. (4.17,63.79) ;
%Curve Lines [id:da7874206506093477]
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (50.96,2.29) .. controls (38.63,31.12) and (38.96,37.12) .. (50.96,63.79) ;
\end{tikzpicture}
=\frac{A^{-1}}{\mathrm{e}^{\mathrm{i} \theta } -A}\tikzset{every picture/.style={line width=0.75pt}} %set default line width to 0.75pt
\begin{tikzpicture}[x=0.75pt,y=0.75pt,yscale=-1,xscale=1, baseline=(XXXX.south) ]
\path (0,67);\path (71.98169708251953,0);\draw ($(current bounding box.center)+(0,0.3em)$) node [anchor=south] (XXXX) {};
%Curve Lines [id:da9101758088105385]
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (68.81,4.31) .. controls (38.96,19.2) and (32.75,18.79) .. (5.15,4.31) ;
%Curve Lines [id:da9071304110135325]
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (68.81,60.79) .. controls (38.96,45.9) and (32.75,46.3) .. (5.15,60.79) ;
\end{tikzpicture}
.
\end{equation*}
At the same time, we also have
\begin{equation*}
\begin{aligned}
\tikzset{every picture/.style={line width=0.75pt}} %set default line width to 0.75pt
\begin{tikzpicture}[x=0.75pt,y=0.75pt,yscale=-1,xscale=1, baseline=(XXXX.south) ]
\path (0,68);\path (68.99652099609375,0);\draw ($(current bounding box.center)+(0,0.3em)$) node [anchor=south] (XXXX) {};
%Straight Lines [id:da7539388624541459]
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (65.7,3.81) -- (3,66.51) ;
%Straight Lines [id:da3938438906428654]
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (41.56,41.59) -- (66.56,66.59) ;
%Straight Lines [id:da18516570828706413]
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (3.23,3.25) -- (28.23,28.25) ;
\end{tikzpicture}
& =A\tikzset{every picture/.style={line width=0.75pt}} %set default line width to 0.75pt
\begin{tikzpicture}[x=0.75pt,y=0.75pt,yscale=-1,xscale=1, baseline=(XXXX.south) ]
\path (0,67);\path (71.98169708251953,0);\draw ($(current bounding box.center)+(0,0.3em)$) node [anchor=south] (XXXX) {};
%Curve Lines [id:da8933911166454194]
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (68.81,4.31) .. controls (38.96,19.2) and (32.75,18.79) .. (5.15,4.31) ;
%Curve Lines [id:da4551593879680247]
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (68.81,60.79) .. controls (38.96,45.9) and (32.75,46.3) .. (5.15,60.79) ;
\end{tikzpicture}
+A^{-1}\tikzset{every picture/.style={line width=0.75pt}} %set default line width to 0.75pt
\begin{tikzpicture}[x=0.75pt,y=0.75pt,yscale=-1,xscale=1, baseline=(XXXX.south) ]
\path (0,68);\path (53.991668701171875,0);\draw ($(current bounding box.center)+(0,0.3em)$) node [anchor=south] (XXXX) {};
%Curve Lines [id:da7648739395234672]
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (4.17,2.29) .. controls (16.5,31.12) and (16.17,37.12) .. (4.17,63.79) ;
%Curve Lines [id:da5044659836989394]
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (50.96,2.29) .. controls (38.63,31.12) and (38.96,37.12) .. (50.96,63.79) ;
\end{tikzpicture}
\\
& \stackrel{!}{=}\mathrm{e}^{-\mathrm{i} \theta }\tikzset{every picture/.style={line width=0.75pt}} %set default line width to 0.75pt
\begin{tikzpicture}[x=0.75pt,y=0.75pt,yscale=-1,xscale=1, baseline=(XXXX.south) ]
\path (0,68);\path (53.991668701171875,0);\draw ($(current bounding box.center)+(0,0.3em)$) node [anchor=south] (XXXX) {};
%Curve Lines [id:da48143120116797333]
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (4.17,2.29) .. controls (16.5,31.12) and (16.17,37.12) .. (4.17,63.79) ;
%Curve Lines [id:da5734280588341469]
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (50.96,2.29) .. controls (38.63,31.12) and (38.96,37.12) .. (50.96,63.79) ;
\end{tikzpicture}
,
\end{aligned}
\end{equation*}
Because exchange two particles in one certain direction gives a phase factor $\mathrm{e}^{\mathrm{i} \theta }$. Then
\begin{equation*}
\tikzset{every picture/.style={line width=0.75pt}} %set default line width to 0.75pt
\begin{tikzpicture}[x=0.75pt,y=0.75pt,yscale=-1,xscale=1, baseline=(XXXX.south) ]
\path (0,68);\path (53.991668701171875,0);\draw ($(current bounding box.center)+(0,0.3em)$) node [anchor=south] (XXXX) {};
%Curve Lines [id:da3082007174180428]
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (4.17,2.29) .. controls (16.5,31.12) and (16.17,37.12) .. (4.17,63.79) ;
%Curve Lines [id:da2760148242111715]
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (50.96,2.29) .. controls (38.63,31.12) and (38.96,37.12) .. (50.96,63.79) ;
\end{tikzpicture}
=\frac{A}{\mathrm{e}^{-\mathrm{i} \theta } -A^{-1}}\tikzset{every picture/.style={line width=0.75pt}} %set default line width to 0.75pt
\begin{tikzpicture}[x=0.75pt,y=0.75pt,yscale=-1,xscale=1, baseline=(XXXX.south) ]
\path (0,67);\path (71.98169708251953,0);\draw ($(current bounding box.center)+(0,0.3em)$) node [anchor=south] (XXXX) {};
%Curve Lines [id:da06451084320784006]
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (68.81,4.31) .. controls (38.96,19.2) and (32.75,18.79) .. (5.15,4.31) ;
%Curve Lines [id:da13449620146904073]
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (68.81,60.79) .. controls (38.96,45.9) and (32.75,46.3) .. (5.15,60.79) ;
\end{tikzpicture}
.
\end{equation*}
So we have the equation
\begin{equation*}
\frac{A}{\mathrm{e}^{-\mathrm{i} \theta } -A^{-1}} =\frac{A^{-1}}{\mathrm{e}^{\mathrm{i} \theta } -A} \Rightarrow \mathrm{e}^{\mathrm{i} \theta } =-A^{3} .
\end{equation*}
By the way, we also have
\begin{equation*}
\tikzset{every picture/.style={line width=0.75pt}} %set default line width to 0.75pt
\begin{tikzpicture}[x=0.75pt,y=0.75pt,yscale=-1,xscale=1, baseline=(XXXX.south) ]
\path (0,68);\path (53.991668701171875,0);\draw ($(current bounding box.center)+(0,0.3em)$) node [anchor=south] (XXXX) {};
%Curve Lines [id:da280533134954952]
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (4.17,2.29) .. controls (16.5,31.12) and (16.17,37.12) .. (4.17,63.79) ;
%Curve Lines [id:da7685079345425179]
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (50.96,2.29) .. controls (38.63,31.12) and (38.96,37.12) .. (50.96,63.79) ;
\end{tikzpicture}
=-\frac{A^{2}}{1+A^{4}}\tikzset{every picture/.style={line width=0.75pt}} %set default line width to 0.75pt
\begin{tikzpicture}[x=0.75pt,y=0.75pt,yscale=-1,xscale=1, baseline=(XXXX.south) ]
\path (0,67);\path (71.98169708251953,0);\draw ($(current bounding box.center)+(0,0.3em)$) node [anchor=south] (XXXX) {};
%Curve Lines [id:da7107080303425743]
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (68.81,4.31) .. controls (38.96,19.2) and (32.75,18.79) .. (5.15,4.31) ;
%Curve Lines [id:da4726019052110213]
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (68.81,60.79) .. controls (38.96,45.9) and (32.75,46.3) .. (5.15,60.79) ;
\end{tikzpicture}
.
\end{equation*}
(a) For $A=\pm \mathrm{e}^{\mathrm{i} \pi /3}$, we have
\begin{equation*}
\mathrm{e}^{\mathrm{i} \theta } =\pm 1,
\end{equation*}
which gives bosons for fermions. And
\begin{equation*}
\tikzset{every picture/.style={line width=0.75pt}} %set default line width to 0.75pt
\begin{tikzpicture}[x=0.75pt,y=0.75pt,yscale=-1,xscale=1, baseline=(XXXX.south) ]
\path (0,68);\path (53.991668701171875,0);\draw ($(current bounding box.center)+(0,0.3em)$) node [anchor=south] (XXXX) {};
%Curve Lines [id:da48157044438828867]
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (4.17,2.29) .. controls (16.5,31.12) and (16.17,37.12) .. (4.17,63.79) ;
%Curve Lines [id:da8477956884783802]
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (50.96,2.29) .. controls (38.63,31.12) and (38.96,37.12) .. (50.96,63.79) ;
\end{tikzpicture}
=\tikzset{every picture/.style={line width=0.75pt}} %set default line width to 0.75pt
\begin{tikzpicture}[x=0.75pt,y=0.75pt,yscale=-1,xscale=1, baseline=(XXXX.south) ]
\path (0,67);\path (71.98169708251953,0);\draw ($(current bounding box.center)+(0,0.3em)$) node [anchor=south] (XXXX) {};
%Curve Lines [id:da6857291636967586]
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (68.81,4.31) .. controls (38.96,19.2) and (32.75,18.79) .. (5.15,4.31) ;
%Curve Lines [id:da8898537612520505]
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (68.81,60.79) .. controls (38.96,45.9) and (32.75,46.3) .. (5.15,60.79) ;
\end{tikzpicture}
,
\end{equation*}
which are fermions and bosons.
(b) For $A=\pm \mathrm{e}^{\mathrm{i} \pi /6}$, we have
\begin{equation*}
\mathrm{e}^{\mathrm{i} \theta } =\mp \mathrm{i} ,
\end{equation*}
which gives bosons for fermions. And
\begin{equation*}
\tikzset{every picture/.style={line width=0.75pt}} %set default line width to 0.75pt
\begin{tikzpicture}[x=0.75pt,y=0.75pt,yscale=-1,xscale=1, baseline=(XXXX.south) ]
\path (0,68);\path (53.991668701171875,0);\draw ($(current bounding box.center)+(0,0.3em)$) node [anchor=south] (XXXX) {};
%Curve Lines [id:da7554397487746631]
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (4.17,2.29) .. controls (16.5,31.12) and (16.17,37.12) .. (4.17,63.79) ;
%Curve Lines [id:da9971387307406114]
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (50.96,2.29) .. controls (38.63,31.12) and (38.96,37.12) .. (50.96,63.79) ;
\end{tikzpicture}
=-\tikzset{every picture/.style={line width=0.75pt}} %set default line width to 0.75pt
\begin{tikzpicture}[x=0.75pt,y=0.75pt,yscale=-1,xscale=1, baseline=(XXXX.south) ]
\path (0,67);\path (71.98169708251953,0);\draw ($(current bounding box.center)+(0,0.3em)$) node [anchor=south] (XXXX) {};
%Curve Lines [id:da5892182617723671]
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (68.81,4.31) .. controls (38.96,19.2) and (32.75,18.79) .. (5.15,4.31) ;
%Curve Lines [id:da7044992807132684]
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (68.81,60.79) .. controls (38.96,45.9) and (32.75,46.3) .. (5.15,60.79) ;
\end{tikzpicture}
,
\end{equation*}
which are semions.
\section{Reidermeister moves and the Kauffman Bracket}
Show that the Kauffman bracket invariant is unchanged under application of Reidermeister move of type II and type III. Thus conclude that the Kauffman invariant is an invariant of regular isotopy.
\paragraph{Answer}
(a) The Reidermeister move of type II is given by Fig.\ref{fig:ReidermeisterMoveOfTypeII}.
\begin{figure}[h!]
\centering
\tikzset{every picture/.style={line width=0.75pt}} %set default line width to 0.75pt
\begin{tikzpicture}[x=0.75pt,y=0.75pt,yscale=-1,xscale=1]
%uncomment if require: \path (0,82); %set diagram left start at 0, and has height of 82
%Curve Lines [id:da28666373624741825]
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (231.17,1.01) .. controls (246.9,37.91) and (246.47,45.59) .. (231.17,79.72) ;
%Curve Lines [id:da6473636293749112]
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (290.85,1.01) .. controls (275.12,37.91) and (275.54,45.59) .. (290.85,79.72) ;
%Curve Lines [id:da34726741926398463]
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (375.31,2.56) .. controls (416.46,20.8) and (409.06,66.44) .. (375.08,77.96) ;
%Curve Lines [id:da9780689263100604]
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (409.7,2.77) .. controls (405.6,4.17) and (401.2,6.73) .. (398.66,8.86) ;
%Curve Lines [id:da16670674465631863]
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (398.43,72.41) .. controls (401.9,74.97) and (406.06,77.1) .. (409.06,78.17) ;
%Curve Lines [id:da3643115225934235]
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (387.56,20.16) .. controls (380.4,28.55) and (378.55,48.1) .. (387.33,60.47) ;
%Straight Lines [id:da4458936620449674]
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (313,40.36) -- (350.7,40.36) ;
\draw [shift={(353.7,40.36)}, rotate = 180] [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (14.21,-4.28) .. controls (9.04,-1.82) and (4.3,-0.39) .. (0,0) .. controls (4.3,0.39) and (9.04,1.82) .. (14.21,4.28) ;
\draw [shift={(310,40.36)}, rotate = 0] [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (14.21,-4.28) .. controls (9.04,-1.82) and (4.3,-0.39) .. (0,0) .. controls (4.3,0.39) and (9.04,1.82) .. (14.21,4.28) ;
\end{tikzpicture}
\caption{Reidermeister move of type II}
\label{fig:ReidermeisterMoveOfTypeII}
\end{figure}
Then according to the definition of the Kauffman bracket, we have:
\begin{equation*}
\begin{aligned}
\tikzset{every picture/.style={line width=0.75pt}} %set default line width to 0.75pt
\begin{tikzpicture}[x=0.75pt,y=0.75pt,yscale=-1,xscale=1, baseline=(XXXX.south) ]
\path (0,83);\path (40.99652099609375,0);\draw ($(current bounding box.center)+(0,0.3em)$) node [anchor=south] (XXXX) {};
%Curve Lines [id:da9604095537894752]
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (3.31,4.56) .. controls (44.46,22.8) and (37.06,68.44) .. (3.08,79.96) ;
%Curve Lines [id:da44353755004889783]
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (37.7,4.77) .. controls (33.6,6.17) and (29.2,8.73) .. (26.66,10.86) ;
%Curve Lines [id:da9884546368227021]
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (26.43,74.41) .. controls (29.9,76.97) and (34.06,79.1) .. (37.06,80.17) ;
%Curve Lines [id:da1642034829302086]
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (15.56,22.16) .. controls (8.4,30.55) and (6.55,50.1) .. (15.33,62.47) ;
\end{tikzpicture}
& =A\tikzset{every picture/.style={line width=0.75pt}} %set default line width to 0.75pt
\begin{tikzpicture}[x=0.75pt,y=0.75pt,yscale=-1,xscale=1, baseline=(XXXX.south) ]
\path (0,68);\path (68.99652099609375,0);\draw ($(current bounding box.center)+(0,0.3em)$) node [anchor=south] (XXXX) {};
%Straight Lines [id:da6104518270284445]
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (65.7,3.81) -- (3,66.51) ;
%Straight Lines [id:da0766348817654059]
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (41.56,41.59) -- (66.56,66.59) ;
%Straight Lines [id:da42798350857778145]
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (3.23,3.25) -- (28.23,28.25) ;
\end{tikzpicture}
+A^{-1}\tikzset{every picture/.style={line width=0.75pt}} %set default line width to 0.75pt
\begin{tikzpicture}[x=0.75pt,y=0.75pt,yscale=-1,xscale=1, baseline=(XXXX.south) ]
\path (0,92);\path (50.99652099609375,0);\draw ($(current bounding box.center)+(0,0.3em)$) node [anchor=south] (XXXX) {};
%Curve Lines [id:da6343061819532925]
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (48.86,3.92) .. controls (27.16,14.74) and (22.64,14.45) .. (2.57,3.92) ;
%Straight Lines [id:da10138287838463445]
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (30.64,71.36) -- (48.82,89.54) ;
%Curve Lines [id:da8125503416637623]
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (20.95,61.67) .. controls (13.06,53.79) and (8.33,48.34) .. (2.78,43.5) .. controls (2.88,15.63) and (47.22,12.36) .. (48.19,43.9) .. controls (39.59,52.34) and (12.69,78.88) .. (2.61,89.48) ;
\end{tikzpicture}
\\
& =A\left( A\tikzset{every picture/.style={line width=0.75pt}} %set default line width to 0.75pt
\begin{tikzpicture}[x=0.75pt,y=0.75pt,yscale=-1,xscale=1, baseline=(XXXX.south) ]
\path (0,67);\path (71.98169708251953,0);\draw ($(current bounding box.center)+(0,0.3em)$) node [anchor=south] (XXXX) {};
%Curve Lines [id:da12181300195230782]
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (68.81,4.31) .. controls (38.96,19.2) and (32.75,18.79) .. (5.15,4.31) ;
%Curve Lines [id:da003208814156384854]
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (68.81,60.79) .. controls (38.96,45.9) and (32.75,46.3) .. (5.15,60.79) ;
\end{tikzpicture}
+A^{-1}\tikzset{every picture/.style={line width=0.75pt}} %set default line width to 0.75pt
\begin{tikzpicture}[x=0.75pt,y=0.75pt,yscale=-1,xscale=1, baseline=(XXXX.south) ]
\path (0,68);\path (53.991668701171875,0);\draw ($(current bounding box.center)+(0,0.3em)$) node [anchor=south] (XXXX) {};
%Curve Lines [id:da296212719001264]
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (4.17,2.29) .. controls (16.5,31.12) and (16.17,37.12) .. (4.17,63.79) ;
%Curve Lines [id:da14704308294878077]
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (50.96,2.29) .. controls (38.63,31.12) and (38.96,37.12) .. (50.96,63.79) ;
\end{tikzpicture}
\right) +A^{-1}\left( A\tikzset{every picture/.style={line width=0.75pt}} %set default line width to 0.75pt
\begin{tikzpicture}[x=0.75pt,y=0.75pt,yscale=-1,xscale=1, baseline=(XXXX.south) ]
\path (0,68);\path (58.99652099609375,0);\draw ($(current bounding box.center)+(0,0.3em)$) node [anchor=south] (XXXX) {};
%Curve Lines [id:da912568481834767]
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (56.68,2.54) .. controls (31.58,15.06) and (26.36,14.72) .. (3.15,2.54) ;
%Curve Lines [id:da17235049610260633]
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (56.68,67.17) .. controls (31.58,54.65) and (26.36,54.99) .. (3.15,67.17) ;
%Shape: Ellipse [id:dp9280843921436426]
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (15.74,34.85) .. controls (15.74,27.02) and (22.08,20.68) .. (29.91,20.68) .. controls (37.74,20.68) and (44.09,27.02) .. (44.09,34.85) .. controls (44.09,42.68) and (37.74,49.03) .. (29.91,49.03) .. controls (22.08,49.03) and (15.74,42.68) .. (15.74,34.85) -- cycle ;
\end{tikzpicture}
+A^{-1}\tikzset{every picture/.style={line width=0.75pt}} %set default line width to 0.75pt
\begin{tikzpicture}[x=0.75pt,y=0.75pt,yscale=-1,xscale=1, baseline=(XXXX.south) ]
\path (0,67);\path (71.98169708251953,0);\draw ($(current bounding box.center)+(0,0.3em)$) node [anchor=south] (XXXX) {};
%Curve Lines [id:da4627840070580258]
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (68.81,4.31) .. controls (38.96,19.2) and (32.75,18.79) .. (5.15,4.31) ;
%Curve Lines [id:da7914769351421946]
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (68.81,60.79) .. controls (38.96,45.9) and (32.75,46.3) .. (5.15,60.79) ;
\end{tikzpicture}
\right)\\
& =A^{2}\tikzset{every picture/.style={line width=0.75pt}} %set default line width to 0.75pt
\begin{tikzpicture}[x=0.75pt,y=0.75pt,yscale=-1,xscale=1, baseline=(XXXX.south) ]
\path (0,67);\path (71.98169708251953,0);\draw ($(current bounding box.center)+(0,0.3em)$) node [anchor=south] (XXXX) {};
%Curve Lines [id:da2751841456474464]
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (68.81,4.31) .. controls (38.96,19.2) and (32.75,18.79) .. (5.15,4.31) ;
%Curve Lines [id:da28927063236750583]
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (68.81,60.79) .. controls (38.96,45.9) and (32.75,46.3) .. (5.15,60.79) ;
\end{tikzpicture}
+\tikzset{every picture/.style={line width=0.75pt}} %set default line width to 0.75pt
\begin{tikzpicture}[x=0.75pt,y=0.75pt,yscale=-1,xscale=1, baseline=(XXXX.south) ]
\path (0,68);\path (53.991668701171875,0);\draw ($(current bounding box.center)+(0,0.3em)$) node [anchor=south] (XXXX) {};
%Curve Lines [id:da32898216042375594]
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (4.17,2.29) .. controls (16.5,31.12) and (16.17,37.12) .. (4.17,63.79) ;
%Curve Lines [id:da2744164658246455]
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (50.96,2.29) .. controls (38.63,31.12) and (38.96,37.12) .. (50.96,63.79) ;
\end{tikzpicture}
+d\tikzset{every picture/.style={line width=0.75pt}} %set default line width to 0.75pt
\begin{tikzpicture}[x=0.75pt,y=0.75pt,yscale=-1,xscale=1, baseline=(XXXX.south) ]
\path (0,67);\path (71.98169708251953,0);\draw ($(current bounding box.center)+(0,0.3em)$) node [anchor=south] (XXXX) {};
%Curve Lines [id:da6870558472651833]
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (68.81,4.31) .. controls (38.96,19.2) and (32.75,18.79) .. (5.15,4.31) ;
%Curve Lines [id:da13357681707506375]
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (68.81,60.79) .. controls (38.96,45.9) and (32.75,46.3) .. (5.15,60.79) ;
\end{tikzpicture}
+A^{-2}\tikzset{every picture/.style={line width=0.75pt}} %set default line width to 0.75pt
\begin{tikzpicture}[x=0.75pt,y=0.75pt,yscale=-1,xscale=1, baseline=(XXXX.south) ]
\path (0,67);\path (71.98169708251953,0);\draw ($(current bounding box.center)+(0,0.3em)$) node [anchor=south] (XXXX) {};
%Curve Lines [id:da8617051673767433]
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (68.81,4.31) .. controls (38.96,19.2) and (32.75,18.79) .. (5.15,4.31) ;
%Curve Lines [id:da5303437875908261]
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (68.81,60.79) .. controls (38.96,45.9) and (32.75,46.3) .. (5.15,60.79) ;
\end{tikzpicture}
\\
& =\tikzset{every picture/.style={line width=0.75pt}} %set default line width to 0.75pt
\begin{tikzpicture}[x=0.75pt,y=0.75pt,yscale=-1,xscale=1, baseline=(XXXX.south) ]
\path (0,68);\path (53.991668701171875,0);\draw ($(current bounding box.center)+(0,0.3em)$) node [anchor=south] (XXXX) {};
%Curve Lines [id:da9892583999632016]
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (4.17,2.29) .. controls (16.5,31.12) and (16.17,37.12) .. (4.17,63.79) ;
%Curve Lines [id:da9019420104276974]
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (50.96,2.29) .. controls (38.63,31.12) and (38.96,37.12) .. (50.96,63.79) ;
\end{tikzpicture}
.
\end{aligned}
\end{equation*}
In the last line we use $d=-A^{2} -A^{-2}$. Now we can see type II is invariant.
(b) The Reidermeister move of type III is given by Fig.\ref{fig:ReidermeisterMoveOfTypeIII}.
\begin{figure}[h!]
\centering
\tikzset{every picture/.style={line width=0.75pt}} %set default line width to 0.75pt
\begin{tikzpicture}[x=0.75pt,y=0.75pt,yscale=-1,xscale=1]
%uncomment if require: \path (0,83); %set diagram left start at 0, and has height of 83
%Straight Lines [id:da11950785204529102]
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (313,40.77) -- (350.7,40.77) ;
\draw [shift={(353.7,40.77)}, rotate = 180] [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (14.21,-4.28) .. controls (9.04,-1.82) and (4.3,-0.39) .. (0,0) .. controls (4.3,0.39) and (9.04,1.82) .. (14.21,4.28) ;
\draw [shift={(310,40.77)}, rotate = 0] [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (14.21,-4.28) .. controls (9.04,-1.82) and (4.3,-0.39) .. (0,0) .. controls (4.3,0.39) and (9.04,1.82) .. (14.21,4.28) ;
%Straight Lines [id:da8207765534905693]
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (229.76,4.84) -- (302.61,77.69) ;
%Straight Lines [id:da43977309937961295]
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (273.65,32.88) -- (302.7,3.84) ;
%Straight Lines [id:da7291965597075933]
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (229.11,77.43) -- (258.16,48.38) ;
%Straight Lines [id:da9127752946580543]
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (251.18,4.66) -- (251.18,19.4) ;
%Straight Lines [id:da5580787674820835]
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (251.18,32.2) -- (251.18,47.51) ;
%Straight Lines [id:da5788807019103526]
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (251.18,61.42) -- (251.18,76.72) ;
%Straight Lines [id:da1822021649397716]
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (359.42,5.16) -- (431.61,77.36) ;
%Straight Lines [id:da719661142881121]
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (398.68,37.19) -- (431.7,4.17) ;
%Straight Lines [id:da0823586599622459]
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (358.78,77.09) -- (391.3,44.58) ;
%Straight Lines [id:da8525034743711262]
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (409.01,4.98) -- (409.01,19.6) ;
%Straight Lines [id:da246160857461047]
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (409.01,32.28) -- (409.01,47.45) ;
%Straight Lines [id:da5380394964109403]
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (409.01,61.23) -- (409.01,76.4) ;
\end{tikzpicture}
\caption{Reidermeister move of type III}
\label{fig:ReidermeisterMoveOfTypeIII}
\end{figure}
Then according to the definition of the Kauffman bracket, we have:
\begin{align*}
\tikzset{every picture/.style={line width=0.75pt}} %set default line width to 0.75pt
\begin{tikzpicture}[x=0.75pt,y=0.75pt,yscale=-1,xscale=1, baseline=(XXXX.south) ]
\path (0,79);\path (76.99652099609375,0);\draw ($(current bounding box.center)+(0,0.3em)$) node [anchor=south] (XXXX) {};
%Straight Lines [id:da5322048377507285]
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (2.76,3.84) -- (75.61,76.69) ;
%Straight Lines [id:da2834370641983299]
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (46.65,31.88) -- (75.7,2.84) ;
%Straight Lines [id:da7335835078220603]
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (2.11,76.43) -- (31.16,47.38) ;
%Straight Lines [id:da6290897514246225]
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (24.18,3.66) -- (24.18,18.4) ;
%Straight Lines [id:da7855778727017275]
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (24.18,31.2) -- (24.18,46.51) ;
%Straight Lines [id:da40285349600704823]
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (24.18,60.42) -- (24.18,75.72) ;
\end{tikzpicture}
& =A\tikzset{every picture/.style={line width=0.75pt}} %set default line width to 0.75pt
\begin{tikzpicture}[x=0.75pt,y=0.75pt,yscale=-1,xscale=1, baseline=(XXXX.south) ]
\path (0,79);\path (97.99652099609375,0);\draw ($(current bounding box.center)+(0,0.3em)$) node [anchor=south] (XXXX) {};
%Straight Lines [id:da651734203263048]
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (22.94,1.73) -- (97.13,75.92) ;
%Straight Lines [id:da9822862230824239]
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (58.66,25.97) -- (82,2.63) ;
%Straight Lines [id:da5738361847067126]
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (8.71,76.73) -- (46.16,39.29) ;
%Straight Lines [id:da651347852864083]
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (37.54,61.48) -- (52.8,76.73) ;
%Straight Lines [id:da16594256373987704]
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (3.89,29.45) -- (22.64,48.2) ;
\end{tikzpicture}
+A^{-1}\tikzset{every picture/.style={line width=0.75pt}} %set default line width to 0.75pt
\begin{tikzpicture}[x=0.75pt,y=0.75pt,yscale=-1,xscale=1, baseline=(XXXX.south) ]
\path (0,81);\path (79.99413299560547,0);\draw ($(current bounding box.center)+(0,0.3em)$) node [anchor=south] (XXXX) {};
%Curve Lines [id:da9862043434565733]
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (62.72,2.31) .. controls (37.09,15.09) and (31.75,14.75) .. (8.04,2.31) ;
%Straight Lines [id:da7232272884844684]
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (47.49,38.48) -- (66.18,19.79) ;
%Straight Lines [id:da5625150004968165]
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (7.51,79.11) -- (37.49,49.14) ;
%Straight Lines [id:da4800775441178744]
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (30.59,66.9) -- (42.81,79.11) ;
%Curve Lines [id:da8772854853652521]
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (18.66,56.27) .. controls (0.27,39.31) and (-2.34,26.91) .. (5.49,21.03) .. controls (13.32,15.16) and (21.8,22.34) .. (78.29,78.46) ;
\end{tikzpicture}
\\
& =A\left( A\tikzset{every picture/.style={line width=0.75pt}} %set default line width to 0.75pt
\begin{tikzpicture}[x=0.75pt,y=0.75pt,yscale=-1,xscale=1, baseline=(XXXX.south) ]
\path (0,74);\path (74.99585723876953,0);\draw ($(current bounding box.center)+(0,0.3em)$) node [anchor=south] (XXXX) {};
%Straight Lines [id:da9591344309737191]
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (4.57,4.74) -- (70.04,48.7) ;
%Straight Lines [id:da22448626545323047]
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (44.02,21.66) -- (70.12,4.14) ;
%Straight Lines [id:da452032535930315]
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (3.99,48.54) -- (30.09,31.01) ;
%Curve Lines [id:da3238575571012392]
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (68.89,69) .. controls (39.04,54.11) and (32.83,54.51) .. (5.22,69) ;
\end{tikzpicture}
+A^{-1}\tikzset{every picture/.style={line width=0.75pt}} %set default line width to 0.75pt
\begin{tikzpicture}[x=0.75pt,y=0.75pt,yscale=-1,xscale=1, baseline=(XXXX.south) ]
\path (0,64);\path (61.991004943847656,0);\draw ($(current bounding box.center)+(0,0.3em)$) node [anchor=south] (XXXX) {};
%Straight Lines [id:da4597054660712325]
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (17.45,4.43) -- (56.2,63.52) ;
%Straight Lines [id:da9036592905897192]
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (40.8,27.18) -- (56.24,3.61) ;
%Straight Lines [id:da16547855392481936]
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (17.11,63.31) -- (32.56,39.74) ;
%Curve Lines [id:da9241917289860497]
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (2.17,2.82) .. controls (14.5,31.65) and (14.17,37.65) .. (2.17,64.32) ;
\end{tikzpicture}
\right) +A^{-1}\left( A\tikzset{every picture/.style={line width=0.75pt}} %set default line width to 0.75pt
\begin{tikzpicture}[x=0.75pt,y=0.75pt,yscale=-1,xscale=1, baseline=(XXXX.south) ]
\path (0,67);\path (65.99652099609375,0);\draw ($(current bounding box.center)+(0,0.3em)$) node [anchor=south] (XXXX) {};
%Curve Lines [id:da3283442283468585]
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (66,2.52) .. controls (36.15,17.41) and (29.94,17) .. (2.33,2.52) ;
%Curve Lines [id:da08588422248637961]
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (66,64) .. controls (36.15,49.11) and (29.94,49.51) .. (2.33,64) ;
%Curve Lines [id:da9555444507529145]
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (40.2,33.6) .. controls (19.2,47.58) and (2.06,47.31) .. (2.13,33.44) .. controls (2.2,19.58) and (50.2,24.6) .. (66.2,41.6) ;
%Curve Lines [id:da2724960587777081]
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (47.2,26.6) .. controls (53.67,22.9) and (58.67,20.23) .. (65.2,20.6) ;
\end{tikzpicture}
+A^{-1}\tikzset{every picture/.style={line width=0.75pt}} %set default line width to 0.75pt
\begin{tikzpicture}[x=0.75pt,y=0.75pt,yscale=-1,xscale=1, baseline=(XXXX.south) ]
\path (0,74);\path (74.99585723876953,0);\draw ($(current bounding box.center)+(0,0.3em)$) node [anchor=south] (XXXX) {};
%Straight Lines [id:da9065450436189566]
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (4.59,20.23) -- (72.71,69.05) ;
%Straight Lines [id:da4804591340207396]
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (45.63,39.02) -- (72.79,19.56) ;
%Straight Lines [id:da3666760694217881]
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (3.99,68.87) -- (31.15,49.4) ;
%Curve Lines [id:da7794924828593213]
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (71.62,5.31) .. controls (40.57,21.84) and (34.1,21.4) .. (5.38,5.31) ;
\end{tikzpicture}
\right)\\
& =A^{2}\tikzset{every picture/.style={line width=0.75pt}} %set default line width to 0.75pt
\begin{tikzpicture}[x=0.75pt,y=0.75pt,yscale=-1,xscale=1, baseline=(XXXX.south) ]
\path (0,74);\path (74.99585723876953,0);\draw ($(current bounding box.center)+(0,0.3em)$) node [anchor=south] (XXXX) {};
%Straight Lines [id:da6841403382603282]
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (4.57,4.74) -- (70.04,48.7) ;
%Straight Lines [id:da8167315388208278]
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (44.02,21.66) -- (70.12,4.14) ;
%Straight Lines [id:da7860921281606539]
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (3.99,48.54) -- (30.09,31.01) ;
%Curve Lines [id:da45872431121640656]
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (68.89,69) .. controls (39.04,54.11) and (32.83,54.51) .. (5.22,69) ;
\end{tikzpicture}
+\tikzset{every picture/.style={line width=0.75pt}} %set default line width to 0.75pt
\begin{tikzpicture}[x=0.75pt,y=0.75pt,yscale=-1,xscale=1, baseline=(XXXX.south) ]
\path (0,64);\path (61.991004943847656,0);\draw ($(current bounding box.center)+(0,0.3em)$) node [anchor=south] (XXXX) {};
%Straight Lines [id:da5762679515989089]
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (17.45,4.43) -- (56.2,63.52) ;
%Straight Lines [id:da5946601466661465]
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (40.8,27.18) -- (56.24,3.61) ;
%Straight Lines [id:da3025487991076059]
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (17.11,63.31) -- (32.56,39.74) ;
%Curve Lines [id:da8120924358614992]
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (2.17,2.82) .. controls (14.5,31.65) and (14.17,37.65) .. (2.17,64.32) ;
\end{tikzpicture}
+\tikzset{every picture/.style={line width=0.75pt}} %set default line width to 0.75pt
\begin{tikzpicture}[x=0.75pt,y=0.75pt,yscale=-1,xscale=1, baseline=(XXXX.south) ]
\path (0,67);\path (65.99652099609375,0);\draw ($(current bounding box.center)+(0,0.3em)$) node [anchor=south] (XXXX) {};
%Curve Lines [id:da8066902275465357]
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (66,2.52) .. controls (36.15,17.41) and (29.94,17) .. (2.33,2.52) ;
%Curve Lines [id:da5803036714274801]
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (66,64) .. controls (36.15,49.11) and (29.94,49.51) .. (2.33,64) ;
%Curve Lines [id:da1465258396067759]
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (40.2,33.6) .. controls (19.2,47.58) and (2.06,47.31) .. (2.13,33.44) .. controls (2.2,19.58) and (50.2,24.6) .. (66.2,41.6) ;
%Curve Lines [id:da7828362623581651]
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (47.2,26.6) .. controls (54.2,21.6) and (58.2,19.6) .. (65.2,20.6) ;
\end{tikzpicture}
+A^{-2}\tikzset{every picture/.style={line width=0.75pt}} %set default line width to 0.75pt
\begin{tikzpicture}[x=0.75pt,y=0.75pt,yscale=-1,xscale=1, baseline=(XXXX.south) ]
\path (0,74);\path (74.99585723876953,0);\draw ($(current bounding box.center)+(0,0.3em)$) node [anchor=south] (XXXX) {};
%Straight Lines [id:da6133566330648923]
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (4.59,20.23) -- (72.71,69.05) ;
%Straight Lines [id:da27516495259545626]
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (45.63,39.02) -- (72.79,19.56) ;
%Straight Lines [id:da44400211184932603]
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (3.99,68.87) -- (31.15,49.4) ;
%Curve Lines [id:da5237433546266892]
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (71.62,5.31) .. controls (40.57,21.84) and (34.1,21.4) .. (5.38,5.31) ;
\end{tikzpicture}
.
\end{align*}
Similarly, we can also work out that
\begin{equation*}
\tikzset{every picture/.style={line width=0.75pt}} %set default line width to 0.75pt
\begin{tikzpicture}[x=0.75pt,y=0.75pt,yscale=-1,xscale=1, baseline=(XXXX.south) ]
\path (0,79);\path (76.99652099609375,0);\draw ($(current bounding box.center)+(0,0.3em)$) node [anchor=south] (XXXX) {};
%Straight Lines [id:da4456215523076539]
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (3.42,6.16) -- (75.61,78.36) ;
%Straight Lines [id:da24465539629543986]
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (42.68,38.19) -- (75.7,5.17) ;
%Straight Lines [id:da21746736537565248]
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (2.78,78.09) -- (35.3,45.58) ;
%Straight Lines [id:da37125615310092086]
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (53.01,5.98) -- (53.01,20.6) ;
%Straight Lines [id:da22315148334215507]
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (53.01,33.28) -- (53.01,48.45) ;
%Straight Lines [id:da13239005571577134]
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (53.01,62.23) -- (53.01,77.4) ;
\end{tikzpicture}
=A^{2}\tikzset{every picture/.style={line width=0.75pt}} %set default line width to 0.75pt
\begin{tikzpicture}[x=0.75pt,y=0.75pt,yscale=-1,xscale=1, baseline=(XXXX.south) ]
\path (0,74);\path (74.99585723876953,0);\draw ($(current bounding box.center)+(0,0.3em)$) node [anchor=south] (XXXX) {};
%Straight Lines [id:da4709299158208502]
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (4.59,20.23) -- (72.71,69.05) ;
%Straight Lines [id:da7368811371103137]
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (45.63,39.02) -- (72.79,19.56) ;
%Straight Lines [id:da9184258204030906]
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (3.99,68.87) -- (31.15,49.4) ;
%Curve Lines [id:da29963768788172396]
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (71.62,5.31) .. controls (40.57,21.84) and (34.1,21.4) .. (5.38,5.31) ;
\end{tikzpicture}
+\tikzset{every picture/.style={line width=0.75pt}} %set default line width to 0.75pt
\begin{tikzpicture}[x=0.75pt,y=0.75pt,yscale=-1,xscale=1, baseline=(XXXX.south) ]
\path (0,67);\path (69.9926528930664,0);\draw ($(current bounding box.center)+(0,0.3em)$) node [anchor=south] (XXXX) {};
%Curve Lines [id:da5933305220930818]
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (2.33,2.52) .. controls (32.18,17.41) and (38.39,17) .. (66,2.52) ;
%Curve Lines [id:da5132826295408173]
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (2.33,64) .. controls (32.18,49.11) and (38.39,49.51) .. (66,64) ;
%Curve Lines [id:da665990401146316]
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (28.13,30.08) .. controls (49.13,16.09) and (66.27,16.37) .. (66.2,30.23) .. controls (66.13,44.09) and (21.58,38.82) .. (2.13,22.08) ;
%Curve Lines [id:da9402553759728991]
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (19.29,36.82) .. controls (15.34,38.56) and (7.58,40.54) .. (3.01,43.96) ;
\end{tikzpicture}
+\tikzset{every picture/.style={line width=0.75pt}} %set default line width to 0.75pt
\begin{tikzpicture}[x=0.75pt,y=0.75pt,yscale=-1,xscale=1, baseline=(XXXX.south) ]
\path (0,66);\path (61.991004943847656,0);\draw ($(current bounding box.center)+(0,0.3em)$) node [anchor=south] (XXXX) {};
%Straight Lines [id:da23107196473023173]
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (40.96,4.43) -- (2.22,63.52) ;
%Straight Lines [id:da23002614456462078]
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (17.62,27.18) -- (2.17,3.61) ;
%Straight Lines [id:da6553195295972762]
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (41.3,63.31) -- (25.85,39.74) ;
%Curve Lines [id:da5022934803705121]
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (56.24,2.82) .. controls (43.91,31.65) and (44.24,37.65) .. (56.24,64.32) ;
\end{tikzpicture}
+A^{-2}\tikzset{every picture/.style={line width=0.75pt}} %set default line width to 0.75pt
\begin{tikzpicture}[x=0.75pt,y=0.75pt,yscale=-1,xscale=1, baseline=(XXXX.south) ]
\path (0,74);\path (74.99585723876953,0);\draw ($(current bounding box.center)+(0,0.3em)$) node [anchor=south] (XXXX) {};
%Straight Lines [id:da38554268799622116]
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (4.57,4.74) -- (70.04,48.7) ;
%Straight Lines [id:da12495382628350904]
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (44.02,21.66) -- (70.12,4.14) ;
%Straight Lines [id:da8665687593626803]
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (3.99,48.54) -- (30.09,31.01) ;
%Curve Lines [id:da07618675458629243]
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (68.89,69) .. controls (39.04,54.11) and (32.83,54.51) .. (5.22,69) ;
\end{tikzpicture}
.
\end{equation*}
If we expand, we will find
\begin{equation*}
\tikzset{every picture/.style={line width=0.75pt}} %set default line width to 0.75pt
\begin{tikzpicture}[x=0.75pt,y=0.75pt,yscale=-1,xscale=1, baseline=(XXXX.south) ]
\path (0,74);\path (74.99585723876953,0);\draw ($(current bounding box.center)+(0,0.3em)$) node [anchor=south] (XXXX) {};
%Straight Lines [id:da13237460150950908]
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (4.59,20.23) -- (72.71,69.05) ;
%Straight Lines [id:da5446181227917517]
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (45.63,39.02) -- (72.79,19.56) ;
%Straight Lines [id:da4076692773124668]
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (3.99,68.87) -- (31.15,49.4) ;
%Curve Lines [id:da785956879155703]
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (71.62,5.31) .. controls (40.57,21.84) and (34.1,21.4) .. (5.38,5.31) ;
\end{tikzpicture}
=\tikzset{every picture/.style={line width=0.75pt}} %set default line width to 0.75pt
\begin{tikzpicture}[x=0.75pt,y=0.75pt,yscale=-1,xscale=1, baseline=(XXXX.south) ]
\path (0,74);\path (74.99585723876953,0);\draw ($(current bounding box.center)+(0,0.3em)$) node [anchor=south] (XXXX) {};
%Straight Lines [id:da1636310386009494]
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (4.57,4.74) -- (70.04,48.7) ;
%Straight Lines [id:da17168608002947305]
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (44.02,21.66) -- (70.12,4.14) ;
%Straight Lines [id:da30205015058719664]
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (3.99,48.54) -- (30.09,31.01) ;
%Curve Lines [id:da4524775843403255]
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (68.89,69) .. controls (39.04,54.11) and (32.83,54.51) .. (5.22,69) ;
\end{tikzpicture}
\end{equation*}
trivially and
\begin{equation*}
\tikzset{every picture/.style={line width=0.75pt}} %set default line width to 0.75pt
\begin{tikzpicture}[x=0.75pt,y=0.75pt,yscale=-1,xscale=1, baseline=(XXXX.south) ]
\path (0,67);\path (65.99652099609375,0);\draw ($(current bounding box.center)+(0,0.3em)$) node [anchor=south] (XXXX) {};
%Curve Lines [id:da6501140268228529]
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (66,2.52) .. controls (36.15,17.41) and (29.94,17) .. (2.33,2.52) ;
%Curve Lines [id:da6914469004290484]
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (66,64) .. controls (36.15,49.11) and (29.94,49.51) .. (2.33,64) ;
%Curve Lines [id:da09499828302410296]
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (40.2,33.6) .. controls (19.2,47.58) and (2.06,47.31) .. (2.13,33.44) .. controls (2.2,19.58) and (50.2,24.6) .. (66.2,41.6) ;
%Curve Lines [id:da6473588123232106]
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (47.2,26.6) .. controls (54.2,21.6) and (58.2,19.6) .. (65.2,20.6) ;
\end{tikzpicture}
=\tikzset{every picture/.style={line width=0.75pt}} %set default line width to 0.75pt
\begin{tikzpicture}[x=0.75pt,y=0.75pt,yscale=-1,xscale=1, baseline=(XXXX.south) ]
\path (0,67);\path (69.9926528930664,0);\draw ($(current bounding box.center)+(0,0.3em)$) node [anchor=south] (XXXX) {};
%Curve Lines [id:da36249838667035195]
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (2.33,2.52) .. controls (32.18,17.41) and (38.39,17) .. (66,2.52) ;
%Curve Lines [id:da40335497785624197]
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (2.33,64) .. controls (32.18,49.11) and (38.39,49.51) .. (66,64) ;
%Curve Lines [id:da22821736423541905]
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (28.13,30.08) .. controls (49.13,16.09) and (66.27,16.37) .. (66.2,30.23) .. controls (66.13,44.09) and (21.58,38.82) .. (2.13,22.08) ;
%Curve Lines [id:da7554157508074169]
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (19.29,36.82) .. controls (15.34,38.56) and (7.58,40.54) .. (3.01,43.96) ;
\end{tikzpicture}
\end{equation*}
using
\begin{equation*}
\tikzset{every picture/.style={line width=0.75pt}} %set default line width to 0.75pt
\begin{tikzpicture}[x=0.75pt,y=0.75pt,yscale=-1,xscale=1, baseline=(XXXX.south) ]
\path (0,92);\path (40.987701416015625,0);\draw ($(current bounding box.center)+(0,0.3em)$) node [anchor=south] (XXXX) {};
%Curve Lines [id:da9306492559485098]
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (6.82,88.45) .. controls (6.7,81.94) and (8.59,58.27) .. (10.19,55.07) ;
%Curve Lines [id:da20313851151588191]
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (14.21,40.44) .. controls (22.99,16.31) and (36.33,29.49) .. (38.33,41.9) .. controls (40.34,54.31) and (33.93,65.52) .. (25.12,65.52) .. controls (16.31,65.52) and (7.5,49.51) .. (6.7,4.26) ;
\end{tikzpicture}
=-A^{3}\tikzset{every picture/.style={line width=0.75pt}} %set default line width to 0.75pt
\begin{tikzpicture}[x=0.75pt,y=0.75pt,yscale=-1,xscale=1, baseline=(XXXX.south) ]
\path (0,72);\path (20.98895263671875,0);\draw ($(current bounding box.center)+(0,0.3em)$) node [anchor=south] (XXXX) {};
%Curve Lines [id:da7909274987223911]
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (16.24,4.82) .. controls (3.91,33.65) and (4.24,39.65) .. (16.24,66.32) ;
\end{tikzpicture}
=\tikzset{every picture/.style={line width=0.75pt}} %set default line width to 0.75pt
\begin{tikzpicture}[x=0.75pt,y=0.75pt,yscale=-1,xscale=1, baseline=(XXXX.south) ]
\path (0,92);\path (40.987701416015625,0);\draw ($(current bounding box.center)+(0,0.3em)$) node [anchor=south] (XXXX) {};
%Curve Lines [id:da708963621643202]
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (36.58,4.26) .. controls (36.7,10.77) and (34.81,34.44) .. (33.21,37.64) ;
%Curve Lines [id:da644659018426573]
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (29.19,52.27) .. controls (20.41,76.41) and (7.07,63.23) .. (5.07,50.81) .. controls (3.07,38.4) and (9.47,27.19) .. (18.28,27.19) .. controls (27.09,27.19) and (35.9,43.21) .. (36.7,88.45) ;
\end{tikzpicture}
.
\end{equation*}
So we have
\begin{equation*}
\tikzset{every picture/.style={line width=0.75pt}} %set default line width to 0.75pt
\begin{tikzpicture}[x=0.75pt,y=0.75pt,yscale=-1,xscale=1, baseline=(XXXX.south) ]
\path (0,79);\path (76.99652099609375,0);\draw ($(current bounding box.center)+(0,0.3em)$) node [anchor=south] (XXXX) {};
%Straight Lines [id:da18892863153575634]
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (2.76,3.84) -- (75.61,76.69) ;
%Straight Lines [id:da9373641828986214]
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (46.65,31.88) -- (75.7,2.84) ;
%Straight Lines [id:da1248941125010763]
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (2.11,76.43) -- (31.16,47.38) ;
%Straight Lines [id:da32650025356983003]
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (24.18,3.66) -- (24.18,18.4) ;
%Straight Lines [id:da6270766169268087]
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (24.18,31.2) -- (24.18,46.51) ;
%Straight Lines [id:da2679094930797157]
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (24.18,60.42) -- (24.18,75.72) ;
\end{tikzpicture}
=\tikzset{every picture/.style={line width=0.75pt}} %set default line width to 0.75pt
\begin{tikzpicture}[x=0.75pt,y=0.75pt,yscale=-1,xscale=1, baseline=(XXXX.south) ]
\path (0,79);\path (76.99652099609375,0);\draw ($(current bounding box.center)+(0,0.3em)$) node [anchor=south] (XXXX) {};
%Straight Lines [id:da27328600813587767]
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (3.42,6.16) -- (75.61,78.36) ;
%Straight Lines [id:da229726884057609]
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (42.68,38.19) -- (75.7,5.17) ;
%Straight Lines [id:da9048312420969273]
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (2.78,78.09) -- (35.3,45.58) ;
%Straight Lines [id:da9358847378647277]
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (53.01,5.98) -- (53.01,20.6) ;
%Straight Lines [id:da8124215341231247]
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (53.01,33.28) -- (53.01,48.45) ;
%Straight Lines [id:da7685527674969999]
\draw [color={rgb, 255:red, 74; green, 144; blue, 226 } ,draw opacity=1 ][line width=1.5] (53.01,62.23) -- (53.01,77.4) ;
\end{tikzpicture}