205 lines
4.2 KiB
TeX
205 lines
4.2 KiB
TeX
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\section{Braids}
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\definition{}
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A \textit{braid} is a set of $n$ strings with fixed ends. Two braids are equivalent if they may be deformed into each other without disconnecting the strings. \par
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Two braids are shown below.
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\begin{center}
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\begin{tikzpicture}
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\pic[rotate=90, name prefix=braid] {
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braid = {s_2^{-1} s_1^{-1} s_2 s_2 s_3^{-1} }
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};
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\braidbars{4}
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\end{tikzpicture}
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\hfill
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\begin{tikzpicture}
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\pic[rotate=90, name prefix=braid] {
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braid = {s_2^{-1} s_3 s_2 s_1 s_1^{-1} s_2^{-1} s_3^{-1} s_1^{-1} s_2 s_2 s_3^{-1} }
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};
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\braidbars{4}
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\end{tikzpicture}
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\end{center}
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\problem{}
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Convince yourself that the braids above are equivalent.
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\vfill
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\pagebreak
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\definition{}
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A braid can be \textit{closed} by conecting its ends:
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\begin{center}
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\begin{tikzpicture}
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\pic[rotate=90, name prefix=braid] {
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braid = {s_2^{-1} s_1^{-1} s_2 s_2 s_3^{-1} }
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};
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\closebraid{4}
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\widebraidbars{4}
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\end{tikzpicture}
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\end{center}
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\problem{}
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When will a closed braid form a knot? \par
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When will a closed braid form a link?
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\vfill
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\problem{}
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Draw a braid that creates a $3$-unlink when closed.
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\vfill
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\pagebreak
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\problem{}<braidify>
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Draw the following knots as closed braids.
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\begin{center}
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\hfill
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\begin{minipage}[t]{0.13\textwidth}
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\begin{center}
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\includegraphics[width=\linewidth]{images/trefoil.png}
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\end{center}
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\end{minipage}
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\hfill
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\begin{minipage}[t]{0.15\textwidth}
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\begin{center}
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\includegraphics[width=\linewidth]{images/closed braid a.png}
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\end{center}
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\end{minipage}
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\hfill
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\begin{minipage}[t]{0.15\textwidth}
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\begin{center}
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\includegraphics[width=\linewidth]{images/closed braid b.png}
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\end{center}
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\end{minipage}
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\hfill~
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\end{center}
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\vfill
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\pagebreak
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\problem{}
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We can describe the projection of a braid by listing which strings cross over and under each other as we move along the braid. \par
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\vspace{2mm}
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For example, consider a three-string braid. If the first string crosses over the second, we'll call that a $1$ crossing. If first string crosses \textbf{under} the second, we'll call that a $-1$ crossing.
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\begin{center}
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\hfill
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\begin{minipage}[t]{0.2\textwidth}
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\begin{center}
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\begin{tikzpicture}
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\pic[
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name prefix = braid,
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braid/number of strands = 3
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] {
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braid = {s_1}
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};
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\end{tikzpicture} \par
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\texttt{1} crossing
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\end{center}
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\end{minipage}
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\hfill
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\begin{minipage}[t]{0.2\textwidth}
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\begin{center}
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\begin{tikzpicture}
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\pic[
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name prefix = braid,
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braid/number of strands = 3
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] {
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braid = {s_1^{-1}}
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};
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\end{tikzpicture} \par
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\texttt{-1} crossing
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\end{center}
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\end{minipage}
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\hfill
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\begin{minipage}[t]{0.2\textwidth}
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\begin{center}
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\begin{tikzpicture}
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\pic[
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name prefix = braid,
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braid/number of strands = 3
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] {
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braid = {s_2}
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};
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\end{tikzpicture} \par
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\texttt{2} crossing
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\end{center}
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\end{minipage}
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\hfill
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\begin{minipage}[t]{0.2\textwidth}
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\begin{center}
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\begin{tikzpicture}
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\pic[
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name prefix = braid,
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braid/number of strands = 3
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] {
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braid = {s_2^{-2}}
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};
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\end{tikzpicture} \par
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\texttt{-2} crossing
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\end{center}
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\end{minipage}
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\hfill~
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\end{center}
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\problem{}
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Verify that the following is a $[1, 2, 1, -2, 1, 2]$ braid. \par
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Read the braid right to left, with the \textbf{bottom} string numbered $1$.
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\begin{center}
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\begin{tikzpicture}
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\pic[
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rotate = 90,
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name prefix = braid,
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braid/number of strands = 3
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] {
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% When we rotate a braid
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braid = {s_1 s_2 s_1 s_2^{-1} s_1 s_2}
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};
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\labelbraidstart{3}
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\end{tikzpicture}
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\end{center}
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\vfill
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\problem{}
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Draw the five-string braid defined by $[1, 3, 4, -3, 2, 4]$
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\begin{solution}
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\begin{center}
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\begin{tikzpicture}
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\pic[
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rotate = 90,
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name prefix = braid,
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braid/number of strands = 5
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] {
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braid = {s_1 s_3 s_4 s_3^{-1} s_2 s_4}
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};
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\labelbraidstart{5}
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\labelbraidend{5}
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\end{tikzpicture}
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\end{center}
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\end{solution}
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\vfill
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\pagebreak
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\problem{}
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Identify the knot generated by the 4-string braid $[(-1, 2, 3)^2,~(3)^3]$ \par
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\hint{$[(1, 2)^2, 3] = [1, 2, 1, 2, 3]$}
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\hint{This knot has 6 crossings. Use the knot table.}
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\vfill
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\problem{}
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Show that the closure of the $n$-string braid $[(1, 2, ..., n-1)^m]$ is a knot iff $m$ and $n$ are coprime.
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\vfill
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\pagebreak
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