From 29147fdaf717d6862b81352dc5bf24f118eb347a Mon Sep 17 00:00:00 2001
From: Maria Marchwicka <maria.marchwicka@gmx.net>
Date: Wed, 27 Nov 2019 21:37:42 +0100
Subject: [PATCH] All corrections for first lecture acording MB tips done
---
lec_04_03.tex | 4 +-
lec_11_03.tex | 12 ++--
lec_20_05.tex | 8 +--
lec_25_02.tex | 107 ++++++++++++++++++++++++------------
lec_mess.tex | 4 +-
lectures_on_knot_theory.tex | 12 ++--
6 files changed, 91 insertions(+), 56 deletions(-)
diff --git a/lec_04_03.tex b/lec_04_03.tex
index d2e89bf..7064a05 100644
--- a/lec_04_03.tex
+++ b/lec_04_03.tex
@@ -183,9 +183,9 @@ $S = \begin{pmatrix}
\Rightarrow \text{trefoil is not trivial.}
\]
\end{example}
-\begin{fact}
+\begin{lemma}
$\Delta_K(t)$ is symmetric.
-\end{fact}
+\end{lemma}
\begin{proof}
Let $S$ be an $n \times n$ matrix.
\begin{align*}
diff --git a/lec_11_03.tex b/lec_11_03.tex
index 4b35f79..1fbddb7 100644
--- a/lec_11_03.tex
+++ b/lec_11_03.tex
@@ -152,21 +152,21 @@ A knot (link) is called alternating if it admits an alternating diagram.
\begin{definition}
A reducible crossing in a knot diagram is a crossing for which we can find a circle such that its intersection with a knot diagram is exactly that crossing. A knot diagram without reducible crossing is called reduced.
\end{definition}
-\begin{fact}
+\begin{lemma}
Any reduced alternating diagram has minimal number of crossings.
-\end{fact}
+\end{lemma}
\begin{definition}
The writhe of the diagram is the difference between the number of positive and negative crossings.
\end{definition}
-\begin{fact}[Tait]
+\begin{lemma}[Tait]
Any two diagrams of the same alternating knot have the same writhe.
-\end{fact}
-\begin{fact}
+\end{lemma}
+\begin{lemma}
An alternating knot has Alexander polynomial of the form:
$
a_1t^{n_1} + a_2t^{n_2} + \dots + a_s t^{n_s}
$, where $n_1 < n_2 < \dots < n_s$ and $a_ia_{i+1} < 0$.
-\end{fact}
+\end{lemma}
\begin{problem}[open]
What is the minimal $\alpha \in \mathbb{R}$ such that if $z$ is a root of the Alexander polynomial of an alternating knot, then $\Re(z) > \alpha$.\\
Remark: alternating knots have very simple knot homologies.
diff --git a/lec_20_05.tex b/lec_20_05.tex
index 70cf05d..543bd13 100644
--- a/lec_20_05.tex
+++ b/lec_20_05.tex
@@ -49,20 +49,20 @@ H_1(\widetilde{X}, \mathbb{Z}[t, t^{-1}]) \times
H_1(\widetilde{X}, \mathbb{Z}[t, t^{-1}]) \longrightarrow \quot{\mathbb{Q}}{\mathbb{Z}[t, t^{-1}]}
\end{align*}
-\begin{fact}
+\begin{lemma}
\begin{align*}
&H_1(\widetilde{X}, \mathbb{Z}[t, t^{-1}]) \cong
\quot{\mathbb{Z}{[t, t^{-1}]}^n}{(tV - V^T)\mathbb{Z}[t, t^{-1}]^n}\;, \\
&\text{where $V$ is a Seifert matrix.}
\end{align*}
-\end{fact}
-\begin{fact}
+\end{lemma}
+\begin{lemma}
\begin{align*}
H_1(\widetilde{X}, \mathbb{Z}[t, t^{-1}]) \times
H_1(\widetilde{X}, \mathbb{Z}[t, t^{-1}]) &\longrightarrow \quot{\mathbb{Q}}{\mathbb{Z}[t, t^{-1}]}\\
(\alpha, \beta) \quad &\mapsto \alpha^{-1}{(t -1)(tV - V^T)}^{-1}\beta
\end{align*}
-\end{fact}
+\end{lemma}
\noindent
Note that $\mathbb{Z}[t, t^{-1}]$ is not PID.
Therefore we don't have primary decomposition of this module.
diff --git a/lec_25_02.tex b/lec_25_02.tex
index 5c7a482..b8d0b81 100644
--- a/lec_25_02.tex
+++ b/lec_25_02.tex
@@ -17,12 +17,12 @@ are shown respectively in
\begin{figure}[h]
\centering
- \begin{subfigure}{0.3\textwidth}
+ \begin{subfigure}{0.45\textwidth}
\centering
\includegraphics[width=0.5\textwidth]
{unknot.png}
\end{subfigure}
- \begin{subfigure}{0.3\textwidth}
+ \begin{subfigure}{0.45\textwidth}
\centering
\includegraphics[width=0.5\textwidth]
{trefoil.png}
@@ -34,22 +34,24 @@ are shown respectively in
\begin{figure}[h]
\centering
- \begin{subfigure}{0.3\textwidth}
+ \begin{subfigure}{0.45\textwidth}
\centering
\includegraphics[width=0.5\textwidth]
{not_injective_knot.png}
\end{subfigure}
- \begin{subfigure}{0.3\textwidth}
+ \begin{subfigure}{0.45\textwidth}
\centering
\includegraphics[width=0.5\textwidth]
{not_smooth_knot.png}
\end{subfigure}
- \caption{Not-knots examples:
- an image of
- a function ${S^1\longrightarrow S^3}$
- that isn't injective (left) and
- of a function
- that isn't smooth (right).}
+ \caption{
+ Not-knots examples:
+ an image of
+ a function ${S^1\longrightarrow S^3}$
+ that is not injective (left) and
+ of a function
+ that is not smooth (right).
+ }
\label{fig:notknot}
\end{figure}
@@ -101,23 +103,40 @@ are shown respectively in
in $S^3$.
\end{definition}
-\begin{example}
-Links:
-\begin{itemize}
-\item
-a trivial link with $3$ components:
-\includegraphics[width=0.2\textwidth]{3unknots.png},
-\item
-a Hopf link: \includegraphics[width=0.13\textwidth]{Hopf.png},
-\item
-a Whitehead link:
-\includegraphics[width=0.13\textwidth]{WhiteheadLink.png},
-\item
-a Borromean link:
-\includegraphics[width=0.1\textwidth]{BorromeanRings.png}.
-\end{itemize}
-\end{example}
-%
+\noindent
+Example of simple links are shown in
+\autoref{fig:links}.
+
+\begin{figure}[h]
+ \centering
+ \begin{subfigure}{0.5\textwidth}
+ \centering
+ \includegraphics[width=1\textwidth]
+ {3unknots.png}
+ \caption{A trivial link with $3$ components.}
+ \end{subfigure}
+ \begin{subfigure}{0.4\textwidth}
+ \centering
+ \includegraphics[width=0.7\textwidth]
+ {Hopf.png}
+ \caption{A Hopf link.}
+ \end{subfigure}
+ \begin{subfigure}{0.4\textwidth}
+ \centering
+ \includegraphics[width=0.8\textwidth]
+ {WhiteheadLink.png},
+ \caption{A Whitehead link.}
+ \end{subfigure}
+ \begin{subfigure}{0.4\textwidth}
+ \centering
+ \includegraphics[width=0.7\textwidth]
+ {BorromeanRings.png}
+ \caption{A Borromean link.}
+ \end{subfigure}
+ \caption{Link examples.}
+ \label{fig:links}
+\end{figure}
+
%
%
\begin{definition}\label{def:link_diagram}
@@ -133,12 +152,24 @@ a Borromean link:
\item there are no triple point.
\end{enumerate}
\end{definition}
+
\noindent
By \Cref{def:link_diagram} the following pictures can not be a part of a diagram:
-\includegraphics[width=0.05\textwidth]{LinkDiagram1.png},
-\includegraphics[width=0.03\textwidth]{LinkDiagram2.png},
-\includegraphics[width=0.05\textwidth]{LinkDiagram3.png}.
-
+\begin{figure}[H]
+ \centering
+ \begin{subfigure}{0.1\textwidth}
+ \includegraphics[width=0.8\textwidth]
+ {LinkDiagram1.png},
+ \end{subfigure}
+ \begin{subfigure}{0.1\textwidth}
+ \includegraphics[width=0.6\textwidth]
+ {LinkDiagram2.png},
+ \end{subfigure}
+ \begin{subfigure}{0.1\textwidth}
+ \includegraphics[width=0.8\textwidth]
+ {LinkDiagram3.png}.
+ \end{subfigure}
+\end{figure}
\noindent
There are under- and overcrossings (tunnels and bridges) on a link diagrams with an obvious meaning.
@@ -152,7 +183,9 @@ We can distinguish two types of crossings: right-handed
$\left(\PICorientpluscross\right)$, called a positive crossing, and left-handed $\left(\PICorientminuscross\right)$, called a negative crossing.
\subsection{Reidemeister moves}
+
A Reidemeister move is one of the three types of operation on a link diagram as shown below:
+
\begin{enumerate}[label=\Roman*]
\item\hfill\\
\includegraphics[width=0.6\textwidth]{rm1.png},
@@ -179,8 +212,10 @@ A Reidemeister move is one of the three types of operation on a link diagram as
%Authors: Arnold, V.I., Varchenko, Alexander, Gusein-Zade, S.M.
\subsection{Seifert surface}
+
\noindent
-Let $D$ be an oriented diagram of a link $L$. We change the diagram by smoothing each crossing:
+Let $D$ be an oriented diagram of a link $L$.
+We change the diagram by smoothing each crossing:
\begin{align*}
\PICorientpluscross \mapsto
\PICorientLRsplit,\\
@@ -192,7 +227,8 @@ a disjoint union of circles on the plane.
Each circle bounds a disks in
$\mathbb{R}^3$
(we choose disks that don't intersect).
- For each smoothed crossing we add a twisted band: right-handed for a positive and left-handed for a negative one.
+For each smoothed crossing we add a twisted band:
+right-handed for a positive and left-handed for a negative one.
We get an orientable surface $\Sigma$
such that $\partial \Sigma = L$.\\
@@ -208,6 +244,7 @@ such that $\partial \Sigma = L$.\\
\end{figure}
\noindent
+
Note: the obtained surface isn't unique and in general doesn't need to be connected, but by taking connected sum of all components we can easily get a connected surface (i.e. we take two disconnected components and cut a disk in each of them: $D_1$ and $D_2$. Then we glue both components on the boundaries: $\partial D_1$ and $\partial D_2$.
\begin{figure}[h]
@@ -312,13 +349,13 @@ Seifert surfaces of minimal genus
\label{fig:unknot}
\end{figure}
-\begin{fact}
+\begin{lemma}
$
g_3(\Sigma) = \frac{1}{2} b_1 (\Sigma) =
\frac{1}{2} \dim_{\mathbb{R}}H_1(\Sigma, \mathbb{R}),
$
where $b_1$ is first Betti number of a surface $\Sigma$.
-\end{fact}
+\end{lemma}
\subsection{Seifert matrix}
Let $L$ be a link and
diff --git a/lec_mess.tex b/lec_mess.tex
index 9541a40..836e4f4 100644
--- a/lec_mess.tex
+++ b/lec_mess.tex
@@ -2,8 +2,8 @@
-\begin{fact}[Milnor Singular Points of Complex Hypersurfaces]
-\end{fact}
+\begin{lemma}[Milnor Singular Points of Complex Hypersurfaces]
+\end{lemma}
%\end{comment}
\noindent
An oriented knot is called negative amphichiral if the mirror image $m(K)$ of $K$ is equivalent the reverse knot of $K$: $K^r$. \\
diff --git a/lectures_on_knot_theory.tex b/lectures_on_knot_theory.tex
index 8728ca3..b07cbb2 100644
--- a/lectures_on_knot_theory.tex
+++ b/lectures_on_knot_theory.tex
@@ -58,14 +58,12 @@
{\newline}{}%
\theoremstyle{break}
-\newtheorem{lemma}{Lemma}[section]
-\newtheorem{fact}{Fact}[section]
-\newtheorem{corollary}{Corollary}[section]
-\newtheorem{proposition}{Proposition}[section]
-\newtheorem{example}{Example}[section]
-\newtheorem{problem}{Problem}[section]
-\newtheorem{definition}{Definition}[section]
\newtheorem{theorem}{Theorem}[section]
+\newtheorem{lemma}[theorem]{Lemma}
+\newtheorem{corollary}[theorem]{Corollary}
+\newtheorem{proposition}[theorem]{Proposition}\newtheorem{example}[theorem]{Example}
+\newtheorem{problem}[theorem]{Problem}
+\newtheorem{definition}[theorem]{Definition}
\newcommand{\contradiction}{%
\ensuremath{{\Rightarrow\mspace{-2mu}\Leftarrow}}}