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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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2
Advanced/Knots/main.tex
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@ -29,7 +29,7 @@
\maketitle \maketitle
<Advanced 2> <Advanced 2>
<Spring 2023> <Spring 2023>
{Knots} {Knots and Braids}
{ {
Prepared by Mark on \today Prepared by Mark on \today
} }

3
Advanced/Knots/parts/1 composition.tex
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@ -80,12 +80,13 @@ Try to make them with a cord. \par
\vfill \vfill
\pagebreak \pagebreak
\definition{} \definition{}
When we compose two knots, we may get different results. To fully understand this fact, we need to define knot \textit{orientation}. When we compose two knots, we may get different results. To fully understand this fact, we need to define knot \textit{orientation}.
\vspace{2mm} \vspace{2mm}
An \textit{orientated knot} is created by defining a \say{direction of travel.} \par An \textit{oriented knot} is created by defining a \say{direction of travel.} \par
There are two distinct ways to compose a pair of oriented knots: There are two distinct ways to compose a pair of oriented knots:
\begin{center} \begin{center}

6
Advanced/Knots/parts/3 sticks.tex
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@ -8,7 +8,7 @@ The \textit{stick number} of a knot is the smallest number of \say{sticks} you m
\end{center} \end{center}
\problem{} \problem{}
Make the trefoil knot with sticks. \par Make a trefoil knot with sticks. \par
How many do you need? How many do you need?
\begin{solution} \begin{solution}
@ -20,11 +20,11 @@ How many do you need?
\vfill \vfill
\problem{} \problem{}
How many sticks will you need to make a figure-eight knot? How many sticks do you need to make a figure-eight knot?
\begin{solution} \begin{solution}
The figure-eight knot has stick number 7. \par The figure-eight knot has stick number 7. \par
In fact, this is the \textit{only} knot with stick number 7. In fact, it is the only knot with stick number 7.
\end{solution} \end{solution}
\vfill \vfill

9
Advanced/Knots/parts/4 braids.tex
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@ -1,8 +1,7 @@
\section{Braids} \section{Braids}
\definition{} \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 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:
Two braids are shown below.
\begin{center} \begin{center}
\begin{tikzpicture} \begin{tikzpicture}
@ -139,7 +138,7 @@ For example, consider a three-string braid. If the first string crosses over the
name prefix = braid, name prefix = braid,
braid/number of strands = 3 braid/number of strands = 3
] { ] {
braid = {s_2^{-2}} braid = {s_2^{-1}}
}; };
\end{tikzpicture} \par \end{tikzpicture} \par
\texttt{-2} crossing \texttt{-2} crossing
@ -150,7 +149,7 @@ For example, consider a three-string braid. If the first string crosses over the
\problem{} \problem{}
Verify that the following is a $[1, 2, 1, -2, 1, 2]$ braid. \par 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{center}
\begin{tikzpicture} \begin{tikzpicture}
@ -199,7 +198,7 @@ Identify the knot generated by the 4-string braid $[(-1, 2, 3)^2,~(3)^3]$ \par
\vfill \vfill
\problem{} \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 \vfill
\pagebreak \pagebreak