From 16f96e3333881c38781f61d1aa29f38ed94c9c39 Mon Sep 17 00:00:00 2001 From: Maria Marchwicka Date: Sun, 2 Jun 2019 17:27:46 +0200 Subject: [PATCH] some inkspace pictures --- lectures_on_knot_theory.tex | 172 ++++++++++++++++++++++++++++++++++-- 1 file changed, 164 insertions(+), 8 deletions(-) diff --git a/lectures_on_knot_theory.tex b/lectures_on_knot_theory.tex index fbb9723..746362f 100644 --- a/lectures_on_knot_theory.tex +++ b/lectures_on_knot_theory.tex @@ -50,6 +50,7 @@ \theoremstyle{break} \newtheorem{lemma}{Lemma} \newtheorem{fact}{Fact} +\newtheorem{corollary}{Corollary} \newtheorem{example}{Example} \newtheorem{definition}{Definition} \newtheorem{theorem}{Theorem} @@ -190,6 +191,12 @@ deformed into each other by a finite sequence of Reidemeister moves (and isotopy %The number of Reidemeister Moves Needed for Unknotting %Joel Hass, Jeffrey C. Lagarias %(Submitted on 2 Jul 1998) +% Piotr Sumata, praca magisterska +% proof - transversality theorem (Thom) + +%Singularities of Differentiable Maps +%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: @@ -198,24 +205,173 @@ Let $D$ be an oriented diagram of a link $L$. We change the diagram by smoothing \PICorientminuscross \mapsto \PICorientLRsplit \end{align*} We smooth all the crossings, so we get 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. We get an orientable surface $\Sigma$ such that $\partial \Sigma = L$.\\ -Note: in general the obtained surface 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 $D_2$; now we glue both components on the boundaries: $\partial D_1$ and $\partial D_2$. \begin{figure}[H] \fontsize{15}{10}\selectfont \centering{ \def\svgwidth{\linewidth} -\resizebox{0.7\textwidth}{!}{\input{images/seifert_surface.pdf_tex}} +\resizebox{0.8\textwidth}{!}{\input{images/seifert_alg.pdf_tex}} \caption{Constructing a Seifert surface.} -\label{fig:surfaceSeifert} +\label{fig:SeifertAlg} } \end{figure} -\includegraphics[width=0.3\textwidth]{seifert3d.png}, +\noindent +Note: in general the obtained surface 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 $D_2$; now we glue both components on the boundaries: $\partial D_1$ and $\partial D_2$. - % transversality theorem -%Thom ? -%Singularities of Differentiable Maps -%Authors: Arnold, V.I., Varchenko, Alexander, Gusein-Zade, S.M. +\begin{figure}[H] +\begin{center} +\includegraphics[width=0.6\textwidth]{seifert_connect.png} +\end{center} +\caption{Connecting two surfaces.} +\label{fig:SeifertConnect} +\end{figure} + +\begin{theorem}[Seifert] +Every link in $S^3$ bounds a surface $\Sigma$ that is compact, connected and orientable. Such a surface is called a Seifert surface. +\end{theorem} +% +\begin{figure}[H] +\fontsize{15}{10}\selectfont +\centering{ +\def\svgwidth{\linewidth} +\resizebox{0.8\textwidth}{!}{\input{images/torus_1_2_3.pdf_tex}} +\caption{Genus of an orientable surface.} +\label{fig:genera} +} +\end{figure} +% +% +\begin{definition} +The three genus $g_3(K)$ ($g(K)$) of a knot $K$ is the minimal genus of a Seifert surface $\Sigma$ for $K$. +\end{definition} + +\begin{corollary} +A knot $K$ is trivial if and only $g_3(K) = 0$. +\end{corollary} + +\noindent +Remark: there are knots that admit non isotopic Seifert surfaces of minimal genus (András Juhász, 2008). + +\begin{definition} +Suppose $\alpha$ and $\beta$ are two simple closed curves in $\mathbb{R}^3$. +On a diagram $L$ consider all crossings between $\alpha$ and $\beta$. Let $N_+$ be the number of positive crossings, $N_-$ - negative. Then the linking number: $lk(\alpha, \beta) = \frac{1}{2}(N_+ - N_-)$. +\end{definition} +\hfill +\\ +Let $\nu(\beta)$ be a tubular neighbourhood of a closed simple curve $\beta$. The linking number can be interpreted via first homology group, where $lk(\alpha, \beta)$ is equal to evaluation of $\alpha$ as element of first homology group in complement of $\beta$ in $S^3$: +\[ +\alpha \in H_1(S^3 \setminus \nu(\beta), \mathbb{Z}) \cong \mathbb{Z}.\] +\begin{example} +\begin{itemize}\hfill +\item +Hopf link\hfill +\begin{figure}[H] +\fontsize{20}{10}\selectfont +\centering{ +\def\svgwidth{\linewidth} +\resizebox{0.4\textwidth}{!}{\input{images/linking_hopf.pdf_tex}} +} +\end{figure} +\item +$T(6, 2)$ link\hfill +\begin{figure}[H] +\fontsize{20}{10}\selectfont +\centering{ +\def\svgwidth{\linewidth} +\resizebox{0.4\textwidth}{!}{\input{images/linking_torus_6_2.pdf_tex}} +} +\end{figure} +\end{itemize} +\end{example} + +Let $L$ be a link and $\Sigma$ be a Seifert surface for $L$. Choose a basis for $H_1(\Sigma, \mathbb{Z})$ consisting of simple closed $\alpha_1, \dots, \alpha_n$. +Let $\alpha_1^+, \dots \alpha_n^+$ be copies of $\alpha_i$ lifted up off the surface. Let $lk(\alpha_i, \alpha_i^+) = \{a_{ij}\}$. Then the matrix $S = \{a_{ij}\}_{i, j =1}^n$ is called a Seifert matrix for $L$. + +\begin{theorem} +The Seifert matrices $S_1$ and $S_2$ for the same link $L$ are S-equivalent, that is, $S_2$ can be obtained from $S_1$ by a sequence of following moves: +\begin{enumerate}[label={(\arabic*)}] +\item +$V \rightarrow AVA^T$ for $A \in $ +\item +$V \rightarrow +\begin{pmatrix} + \alpha & * \\ + \gamma^{*} & \delta +\end{pmatrix} +$\\ +\[ + \begin{pmatrix} + \begin{array}{c|c} + \epsilon' [T|_A]\epsilon & \ast \\ + \hline + 0 & _{\overline{B}'} [\overline{T}] + _{\overline{B}\vphantom{\overline{B}'}} + \end{array} + \end{pmatrix} +\]\\ +\[\left| +\begin{array}{cr} + Q & \begin{matrix} 0 \\ 0 \end{matrix} \\ + \begin{matrix} 2 & 3 \end{matrix} & -1 +\end{array} +\right|\] +\\ +\[ +\left[ +\begin{array}{c@{}c@{}c} + \left[\begin{array}{cc} + a_{11} & a_{12} \\ + a_{21} & a_{22} \\ + \end{array}\right] & \mathbf{0} & \mathbf{0} \\ + \mathbf{0} & \left[\begin{array}{ccc} + b_{11} & b_{12} & b_{13}\\ + b_{21} & b_{22} & b_{23}\\ + b_{31} & b_{32} & b_{33}\\ + \end{array}\right] & \mathbf{0}\\ +\mathbf{0} & \mathbf{0} & \left[ \begin{array}{cc} +c_{11} & c_{12} \\ +c_{21} & c_{22} \\ +\end{array}\right] \\ +\end{array}\right] +\] \\ +\[ +\begin{bmatrix} + \begin{bmatrix} + a_{11} & a_{12}\\ + a_{21} & a_{22}\\ + \end{bmatrix} & \mathbf{0} & \mathbf{0} \\ + \mathbf{0} & \begin{bmatrix} + b_{11} & b_{12} & b_{13}\\ + b_{21} & b_{22} & b_{23}\\ + b_{31} & b_{32} & b_{33}\\ + \end{bmatrix} & \mathbf{0} \\ + \mathbf{0} & \mathbf{0} & \begin{bmatrix} + c_{11} & c_{12}\\ + c_{21} & c_{22}\\ + \end{bmatrix} \\ +\end{bmatrix} +\]\\ +\setlength{\arraycolsep}{2em} +\newcommand{\lbrce}{\smash{\left.\rule{0pt}{25pt}\right\}}} +\newcommand{\rbrce}{\smash{\left\{\rule{0pt}{25pt}\right.}} +\newcommand{\sdots}{\smash{\vdots}} +\[ + \begin{pmatrix} + 0 & 0 & 0 \\ + \sdots & \sdots\makebox[0pt][l]{$\lbrce\left\lceil\frac i2\right\rceil$} & \sdots \\ + 0 & 0 & \\ + & & 0 \\ + & & \makebox[0pt][r]{$\left\lfloor\frac i2\right\rfloor\rbrce$}\sdots \\ + 0 & & 0 + \end{pmatrix} +\] + +\item +inverse of (2) + +\end{enumerate} +\end{theorem} \section{} \begin{example}