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Mark
2023-05-05 15:06:30 -07:00
parent 19335875e4
commit fad0f5b896
4 changed files with 10 additions and 10 deletions
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9
Advanced/Knots/parts/4 braids.tex
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@ -1,8 +1,7 @@
\section{Braids}
\definition{}
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
Two braids are shown below.
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. Two braids are shown below:
\begin{center}
\begin{tikzpicture}
@ -139,7 +138,7 @@ For example, consider a three-string braid. If the first string crosses over the
name prefix = braid,
braid/number of strands = 3
] {
braid = {s_2^{-2}}
braid = {s_2^{-1}}
};
\end{tikzpicture} \par
\texttt{-2} crossing
@ -150,7 +149,7 @@ For example, consider a three-string braid. If the first string crosses over the
\problem{}
Verify that the following is a $[1, 2, 1, -2, 1, 2]$ braid. \par
Read the braid right to left, with the \textbf{bottom} string numbered $1$.
Read the braid left to right, with the bottom string numbered $1$.
\begin{center}
\begin{tikzpicture}
@ -199,7 +198,7 @@ Identify the knot generated by the 4-string braid $[(-1, 2, 3)^2,~(3)^3]$ \par
\vfill
\problem{}
Show that the closure of the $n$-string braid $[(1, 2, ..., n-1)^m]$ is a knot iff $m$ and $n$ are coprime.
Show that the $n$-string braid $[(1, 2, ..., n-1)^m]$ forms a knot iff $m$ and $n$ are coprime.
\vfill
\pagebreak