From fad0f5b896c538531aac57533ea9486db4b84b9e Mon Sep 17 00:00:00 2001 From: Mark Date: Fri, 5 May 2023 15:06:30 -0700 Subject: [PATCH] Cleanup --- Advanced/Knots/main.tex | 2 +- Advanced/Knots/parts/1 composition.tex | 3 ++- Advanced/Knots/parts/3 sticks.tex | 6 +++--- Advanced/Knots/parts/4 braids.tex | 9 ++++----- 4 files changed, 10 insertions(+), 10 deletions(-) diff --git a/Advanced/Knots/main.tex b/Advanced/Knots/main.tex index 26a7a16..0a1d0c9 100755 --- a/Advanced/Knots/main.tex +++ b/Advanced/Knots/main.tex @@ -29,7 +29,7 @@ \maketitle - {Knots} + {Knots and Braids} { Prepared by Mark on \today } diff --git a/Advanced/Knots/parts/1 composition.tex b/Advanced/Knots/parts/1 composition.tex index c3b4341..e20e1db 100644 --- a/Advanced/Knots/parts/1 composition.tex +++ b/Advanced/Knots/parts/1 composition.tex @@ -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} diff --git a/Advanced/Knots/parts/3 sticks.tex b/Advanced/Knots/parts/3 sticks.tex index ff62b9c..d47a6a2 100644 --- a/Advanced/Knots/parts/3 sticks.tex +++ b/Advanced/Knots/parts/3 sticks.tex @@ -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 diff --git a/Advanced/Knots/parts/4 braids.tex b/Advanced/Knots/parts/4 braids.tex index 9947dd4..6f43f7c 100644 --- a/Advanced/Knots/parts/4 braids.tex +++ b/Advanced/Knots/parts/4 braids.tex @@ -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 \ No newline at end of file