All corrections for first lecture acording MB tips done

This commit is contained in:
Maria Marchwicka 2019-11-27 21:37:42 +01:00
parent 60d7f7d850
commit 29147fdaf7
6 changed files with 91 additions and 56 deletions

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@ -183,9 +183,9 @@ $S = \begin{pmatrix}
\Rightarrow \text{trefoil is not trivial.} \Rightarrow \text{trefoil is not trivial.}
\] \]
\end{example} \end{example}
\begin{fact} \begin{lemma}
$\Delta_K(t)$ is symmetric. $\Delta_K(t)$ is symmetric.
\end{fact} \end{lemma}
\begin{proof} \begin{proof}
Let $S$ be an $n \times n$ matrix. Let $S$ be an $n \times n$ matrix.
\begin{align*} \begin{align*}

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@ -152,21 +152,21 @@ A knot (link) is called alternating if it admits an alternating diagram.
\begin{definition} \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. 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} \end{definition}
\begin{fact} \begin{lemma}
Any reduced alternating diagram has minimal number of crossings. Any reduced alternating diagram has minimal number of crossings.
\end{fact} \end{lemma}
\begin{definition} \begin{definition}
The writhe of the diagram is the difference between the number of positive and negative crossings. The writhe of the diagram is the difference between the number of positive and negative crossings.
\end{definition} \end{definition}
\begin{fact}[Tait] \begin{lemma}[Tait]
Any two diagrams of the same alternating knot have the same writhe. Any two diagrams of the same alternating knot have the same writhe.
\end{fact} \end{lemma}
\begin{fact} \begin{lemma}
An alternating knot has Alexander polynomial of the form: An alternating knot has Alexander polynomial of the form:
$ $
a_1t^{n_1} + a_2t^{n_2} + \dots + a_s t^{n_s} a_1t^{n_1} + a_2t^{n_2} + \dots + a_s t^{n_s}
$, where $n_1 < n_2 < \dots < n_s$ and $a_ia_{i+1} < 0$. $, where $n_1 < n_2 < \dots < n_s$ and $a_ia_{i+1} < 0$.
\end{fact} \end{lemma}
\begin{problem}[open] \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 $\Re(z) > \alpha$.\\ 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. Remark: alternating knots have very simple knot homologies.

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@ -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}]} H_1(\widetilde{X}, \mathbb{Z}[t, t^{-1}]) \longrightarrow \quot{\mathbb{Q}}{\mathbb{Z}[t, t^{-1}]}
\end{align*} \end{align*}
\begin{fact} \begin{lemma}
\begin{align*} \begin{align*}
&H_1(\widetilde{X}, \mathbb{Z}[t, t^{-1}]) \cong &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}\;, \\ \quot{\mathbb{Z}{[t, t^{-1}]}^n}{(tV - V^T)\mathbb{Z}[t, t^{-1}]^n}\;, \\
&\text{where $V$ is a Seifert matrix.} &\text{where $V$ is a Seifert matrix.}
\end{align*} \end{align*}
\end{fact} \end{lemma}
\begin{fact} \begin{lemma}
\begin{align*} \begin{align*}
H_1(\widetilde{X}, \mathbb{Z}[t, t^{-1}]) \times 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}]}\\ 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 (\alpha, \beta) \quad &\mapsto \alpha^{-1}{(t -1)(tV - V^T)}^{-1}\beta
\end{align*} \end{align*}
\end{fact} \end{lemma}
\noindent \noindent
Note that $\mathbb{Z}[t, t^{-1}]$ is not PID. Note that $\mathbb{Z}[t, t^{-1}]$ is not PID.
Therefore we don't have primary decomposition of this module. Therefore we don't have primary decomposition of this module.

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@ -17,12 +17,12 @@ are shown respectively in
\begin{figure}[h] \begin{figure}[h]
\centering \centering
\begin{subfigure}{0.3\textwidth} \begin{subfigure}{0.45\textwidth}
\centering \centering
\includegraphics[width=0.5\textwidth] \includegraphics[width=0.5\textwidth]
{unknot.png} {unknot.png}
\end{subfigure} \end{subfigure}
\begin{subfigure}{0.3\textwidth} \begin{subfigure}{0.45\textwidth}
\centering \centering
\includegraphics[width=0.5\textwidth] \includegraphics[width=0.5\textwidth]
{trefoil.png} {trefoil.png}
@ -34,22 +34,24 @@ are shown respectively in
\begin{figure}[h] \begin{figure}[h]
\centering \centering
\begin{subfigure}{0.3\textwidth} \begin{subfigure}{0.45\textwidth}
\centering \centering
\includegraphics[width=0.5\textwidth] \includegraphics[width=0.5\textwidth]
{not_injective_knot.png} {not_injective_knot.png}
\end{subfigure} \end{subfigure}
\begin{subfigure}{0.3\textwidth} \begin{subfigure}{0.45\textwidth}
\centering \centering
\includegraphics[width=0.5\textwidth] \includegraphics[width=0.5\textwidth]
{not_smooth_knot.png} {not_smooth_knot.png}
\end{subfigure} \end{subfigure}
\caption{Not-knots examples: \caption{
Not-knots examples:
an image of an image of
a function ${S^1\longrightarrow S^3}$ a function ${S^1\longrightarrow S^3}$
that isn't injective (left) and that is not injective (left) and
of a function of a function
that isn't smooth (right).} that is not smooth (right).
}
\label{fig:notknot} \label{fig:notknot}
\end{figure} \end{figure}
@ -101,23 +103,40 @@ are shown respectively in
in $S^3$. in $S^3$.
\end{definition} \end{definition}
\begin{example} \noindent
Links: Example of simple links are shown in
\begin{itemize} \autoref{fig:links}.
\item
a trivial link with $3$ components: \begin{figure}[h]
\includegraphics[width=0.2\textwidth]{3unknots.png}, \centering
\item \begin{subfigure}{0.5\textwidth}
a Hopf link: \includegraphics[width=0.13\textwidth]{Hopf.png}, \centering
\item \includegraphics[width=1\textwidth]
a Whitehead link: {3unknots.png}
\includegraphics[width=0.13\textwidth]{WhiteheadLink.png}, \caption{A trivial link with $3$ components.}
\item \end{subfigure}
a Borromean link: \begin{subfigure}{0.4\textwidth}
\includegraphics[width=0.1\textwidth]{BorromeanRings.png}. \centering
\end{itemize} \includegraphics[width=0.7\textwidth]
\end{example} {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} \begin{definition}\label{def:link_diagram}
@ -133,12 +152,24 @@ a Borromean link:
\item there are no triple point. \item there are no triple point.
\end{enumerate} \end{enumerate}
\end{definition} \end{definition}
\noindent \noindent
By \Cref{def:link_diagram} the following pictures can not be a part of a diagram: By \Cref{def:link_diagram} the following pictures can not be a part of a diagram:
\includegraphics[width=0.05\textwidth]{LinkDiagram1.png}, \begin{figure}[H]
\includegraphics[width=0.03\textwidth]{LinkDiagram2.png}, \centering
\includegraphics[width=0.05\textwidth]{LinkDiagram3.png}. \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 \noindent
There are under- and overcrossings (tunnels and bridges) on a link diagrams with an obvious meaning. 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. $\left(\PICorientpluscross\right)$, called a positive crossing, and left-handed $\left(\PICorientminuscross\right)$, called a negative crossing.
\subsection{Reidemeister moves} \subsection{Reidemeister moves}
A Reidemeister move is one of the three types of operation on a link diagram as shown below: A Reidemeister move is one of the three types of operation on a link diagram as shown below:
\begin{enumerate}[label=\Roman*] \begin{enumerate}[label=\Roman*]
\item\hfill\\ \item\hfill\\
\includegraphics[width=0.6\textwidth]{rm1.png}, \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. %Authors: Arnold, V.I., Varchenko, Alexander, Gusein-Zade, S.M.
\subsection{Seifert surface} \subsection{Seifert surface}
\noindent \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*} \begin{align*}
\PICorientpluscross \mapsto \PICorientpluscross \mapsto
\PICorientLRsplit,\\ \PICorientLRsplit,\\
@ -192,7 +227,8 @@ a disjoint union of circles on the plane.
Each circle bounds a disks in Each circle bounds a disks in
$\mathbb{R}^3$ $\mathbb{R}^3$
(we choose disks that don't intersect). (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 $\Sigma$ We get an orientable surface $\Sigma$
such that $\partial \Sigma = L$.\\ such that $\partial \Sigma = L$.\\
@ -208,6 +244,7 @@ such that $\partial \Sigma = L$.\\
\end{figure} \end{figure}
\noindent \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 $\partial D_2$. 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] \begin{figure}[h]
@ -312,13 +349,13 @@ Seifert surfaces of minimal genus
\label{fig:unknot} \label{fig:unknot}
\end{figure} \end{figure}
\begin{fact} \begin{lemma}
$ $
g_3(\Sigma) = \frac{1}{2} b_1 (\Sigma) = g_3(\Sigma) = \frac{1}{2} b_1 (\Sigma) =
\frac{1}{2} \dim_{\mathbb{R}}H_1(\Sigma, \mathbb{R}), \frac{1}{2} \dim_{\mathbb{R}}H_1(\Sigma, \mathbb{R}),
$ $
where $b_1$ is first Betti number of a surface $\Sigma$. where $b_1$ is first Betti number of a surface $\Sigma$.
\end{fact} \end{lemma}
\subsection{Seifert matrix} \subsection{Seifert matrix}
Let $L$ be a link and Let $L$ be a link and

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@ -2,8 +2,8 @@
\begin{fact}[Milnor Singular Points of Complex Hypersurfaces] \begin{lemma}[Milnor Singular Points of Complex Hypersurfaces]
\end{fact} \end{lemma}
%\end{comment} %\end{comment}
\noindent \noindent
An oriented knot is called negative amphichiral if the mirror image $m(K)$ of $K$ is equivalent the reverse knot of $K$: $K^r$. \\ An oriented knot is called negative amphichiral if the mirror image $m(K)$ of $K$ is equivalent the reverse knot of $K$: $K^r$. \\

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@ -58,14 +58,12 @@
{\newline}{}% {\newline}{}%
\theoremstyle{break} \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{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}{% \newcommand{\contradiction}{%
\ensuremath{{\Rightarrow\mspace{-2mu}\Leftarrow}}} \ensuremath{{\Rightarrow\mspace{-2mu}\Leftarrow}}}