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Mark 2023-05-05 15:06:30 -07:00
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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
<Advanced 2>
<Spring 2023>
{Knots}
{Knots and Braids}
{
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
\pagebreak
\definition{}
When we compose two knots, we may get different results. To fully understand this fact, we need to define knot \textit{orientation}.
\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:
\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}
\problem{}
Make the trefoil knot with sticks. \par
Make a trefoil knot with sticks. \par
How many do you need?
\begin{solution}
@ -20,11 +20,11 @@ How many do you need?
\vfill
\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}
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}
\vfill

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