some inkspace pictures

2019-06-02 17:27:46 +02:00 · 2019-06-02 17:27:46 +02:00 · 16f96e3333
parent 8574646027
commit 16f96e3333
1 changed files with 164 additions and 8 deletions
--- 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}