some inkspace pictures

images/ball_4.pdf_tex

@@ -0,0 +1,63 @@
+%% Creator: Inkscape inkscape 0.91, www.inkscape.org
+%% PDF/EPS/PS + LaTeX output extension by Johan Engelen, 2010
+%% Accompanies image file 'ball_4.pdf' (pdf, eps, ps)
+%%
+%% To include the image in your LaTeX document, write
+%%   \input{<filename>.pdf_tex}
+%%  instead of
+%%   \includegraphics{<filename>.pdf}
+%% To scale the image, write
+%%   \def\svgwidth{<desired width>}
+%%   \input{<filename>.pdf_tex}
+%%  instead of
+%%   \includegraphics[width=<desired width>]{<filename>.pdf}
+%%
+%% Images with a different path to the parent latex file can
+%% be accessed with the `import' package (which may need to be
+%% installed) using
+%%   \usepackage{import}
+%% in the preamble, and then including the image with
+%%   \import{<path to file>}{<filename>.pdf_tex}
+%% Alternatively, one can specify
+%%   \graphicspath{{<path to file>/}}
+%% 
+%% For more information, please see info/svg-inkscape on CTAN:
+%%   http://tug.ctan.org/tex-archive/info/svg-inkscape
+%%
+\begingroup%
+  \makeatletter%
+  \providecommand\color[2][]{%
+    \errmessage{(Inkscape) Color is used for the text in Inkscape, but the package 'color.sty' is not loaded}%
+    \renewcommand\color[2][]{}%
+  }%
+  \providecommand\transparent[1]{%
+    \errmessage{(Inkscape) Transparency is used (non-zero) for the text in Inkscape, but the package 'transparent.sty' is not loaded}%
+    \renewcommand\transparent[1]{}%
+  }%
+  \providecommand\rotatebox[2]{#2}%
+  \ifx\svgwidth\undefined%
+    \setlength{\unitlength}{538.34058867bp}%
+    \ifx\svgscale\undefined%
+      \relax%
+    \else%
+      \setlength{\unitlength}{\unitlength * \real{\svgscale}}%
+    \fi%
+  \else%
+    \setlength{\unitlength}{\svgwidth}%
+  \fi%
+  \global\let\svgwidth\undefined%
+  \global\let\svgscale\undefined%
+  \makeatother%
+  \begin{picture}(1,0.36148366)%
+    \put(0,0){\includegraphics[width=\unitlength,page=1]{ball_4.pdf}}%
+    \put(0.62288514,0.24829785){\color[rgb]{0,0,0}\makebox(0,0)[lt]{\begin{minipage}{0.62747874\unitlength}\raggedright $B^4$\\ \end{minipage}}}%
+    \put(1.24285327,1.51222577){\color[rgb]{0,0,0}\makebox(0,0)[lt]{\begin{minipage}{0.88153562\unitlength}\raggedright \end{minipage}}}%
+    \put(0,0){\includegraphics[width=\unitlength,page=2]{ball_4.pdf}}%
+    \put(0.08156744,0.26027166){\color[rgb]{0,0,0}\makebox(0,0)[lt]{\begin{minipage}{0.62747874\unitlength}\raggedright $B^4$\\ \end{minipage}}}%
+    \put(0,0){\includegraphics[width=\unitlength,page=3]{ball_4.pdf}}%
+    \put(0.72054873,0.78436255){\color[rgb]{0,0,0}\makebox(0,0)[lt]{\begin{minipage}{0.17856087\unitlength}\raggedright \end{minipage}}}%
+    \put(0.34790192,0.21447686){\color[rgb]{0,0,0}\makebox(0,0)[lt]{\begin{minipage}{0.17197199\unitlength}\raggedright $\Sigma$\end{minipage}}}%
+    \put(0.86664402,0.76488318){\color[rgb]{0,0,0}\makebox(0,0)[lb]{\smash{}}}%
+    \put(0.88802542,0.20006421){\color[rgb]{0,0,0}\makebox(0,0)[lt]{\begin{minipage}{0.17197199\unitlength}\raggedright $\widetilde{\Sigma}$\end{minipage}}}%
+  \end{picture}%
+\endgroup%

images/ink_diag.pdf_tex

@@ -0,0 +1,58 @@
+%% Creator: Inkscape inkscape 0.91, www.inkscape.org
+%% PDF/EPS/PS + LaTeX output extension by Johan Engelen, 2010
+%% Accompanies image file 'ink_diag.pdf' (pdf, eps, ps)
+%%
+%% To include the image in your LaTeX document, write
+%%   \input{<filename>.pdf_tex}
+%%  instead of
+%%   \includegraphics{<filename>.pdf}
+%% To scale the image, write
+%%   \def\svgwidth{<desired width>}
+%%   \input{<filename>.pdf_tex}
+%%  instead of
+%%   \includegraphics[width=<desired width>]{<filename>.pdf}
+%%
+%% Images with a different path to the parent latex file can
+%% be accessed with the `import' package (which may need to be
+%% installed) using
+%%   \usepackage{import}
+%% in the preamble, and then including the image with
+%%   \import{<path to file>}{<filename>.pdf_tex}
+%% Alternatively, one can specify
+%%   \graphicspath{{<path to file>/}}
+%% 
+%% For more information, please see info/svg-inkscape on CTAN:
+%%   http://tug.ctan.org/tex-archive/info/svg-inkscape
+%%
+\begingroup%
+  \makeatletter%
+  \providecommand\color[2][]{%
+    \errmessage{(Inkscape) Color is used for the text in Inkscape, but the package 'color.sty' is not loaded}%
+    \renewcommand\color[2][]{}%
+  }%
+  \providecommand\transparent[1]{%
+    \errmessage{(Inkscape) Transparency is used (non-zero) for the text in Inkscape, but the package 'transparent.sty' is not loaded}%
+    \renewcommand\transparent[1]{}%
+  }%
+  \providecommand\rotatebox[2]{#2}%
+  \ifx\svgwidth\undefined%
+    \setlength{\unitlength}{595.27559055bp}%
+    \ifx\svgscale\undefined%
+      \relax%
+    \else%
+      \setlength{\unitlength}{\unitlength * \real{\svgscale}}%
+    \fi%
+  \else%
+    \setlength{\unitlength}{\svgwidth}%
+  \fi%
+  \global\let\svgwidth\undefined%
+  \global\let\svgscale\undefined%
+  \makeatother%
+  \begin{picture}(1,1.41428571)%
+    \put(0,0){\includegraphics[width=\unitlength,page=1]{ink_diag.pdf}}%
+    \put(0.63682602,0.48349309){\color[rgb]{0,0,0}\makebox(0,0)[lb]{\smash{$B^4$}}}%
+    \put(1.19167986,1.92661837){\color[rgb]{0,0,0}\makebox(0,0)[lt]{\begin{minipage}{0.79722134\unitlength}\raggedright \end{minipage}}}%
+    \put(0.35293654,0.78061261){\color[rgb]{0,0,0}\makebox(0,0)[lt]{\begin{minipage}{0.46919801\unitlength}\raggedright \end{minipage}}}%
+    \put(0.26158827,0.80552578){\color[rgb]{0,0,0}\makebox(0,0)[lt]{\begin{minipage}{0.48165458\unitlength}\raggedright \end{minipage}}}%
+  \end{picture}%
+\endgroup%

images/link_diagram.pdf_tex

@@ -0,0 +1,62 @@
+%% Creator: Inkscape inkscape 0.91, www.inkscape.org
+%% PDF/EPS/PS + LaTeX output extension by Johan Engelen, 2010
+%% Accompanies image file 'link_diagram.pdf' (pdf, eps, ps)
+%%
+%% To include the image in your LaTeX document, write
+%%   \input{<filename>.pdf_tex}
+%%  instead of
+%%   \includegraphics{<filename>.pdf}
+%% To scale the image, write
+%%   \def\svgwidth{<desired width>}
+%%   \input{<filename>.pdf_tex}
+%%  instead of
+%%   \includegraphics[width=<desired width>]{<filename>.pdf}
+%%
+%% Images with a different path to the parent latex file can
+%% be accessed with the `import' package (which may need to be
+%% installed) using
+%%   \usepackage{import}
+%% in the preamble, and then including the image with
+%%   \import{<path to file>}{<filename>.pdf_tex}
+%% Alternatively, one can specify
+%%   \graphicspath{{<path to file>/}}
+%% 
+%% For more information, please see info/svg-inkscape on CTAN:
+%%   http://tug.ctan.org/tex-archive/info/svg-inkscape
+%%
+\begingroup%
+  \makeatletter%
+  \providecommand\color[2][]{%
+    \errmessage{(Inkscape) Color is used for the text in Inkscape, but the package 'color.sty' is not loaded}%
+    \renewcommand\color[2][]{}%
+  }%
+  \providecommand\transparent[1]{%
+    \errmessage{(Inkscape) Transparency is used (non-zero) for the text in Inkscape, but the package 'transparent.sty' is not loaded}%
+    \renewcommand\transparent[1]{}%
+  }%
+  \providecommand\rotatebox[2]{#2}%
+  \ifx\svgwidth\undefined%
+    \setlength{\unitlength}{595.27559055bp}%
+    \ifx\svgscale\undefined%
+      \relax%
+    \else%
+      \setlength{\unitlength}{\unitlength * \real{\svgscale}}%
+    \fi%
+  \else%
+    \setlength{\unitlength}{\svgwidth}%
+  \fi%
+  \global\let\svgwidth\undefined%
+  \global\let\svgscale\undefined%
+  \makeatother%
+  \begin{picture}(1,1.41428571)%
+    \put(0,0){\includegraphics[width=\unitlength,page=1]{link_diagram.pdf}}%
+    \put(0.31120017,0.7509725){\color[rgb]{0,0,0}\makebox(0,0)[lt]{\begin{minipage}{0.77132343\unitlength}\raggedright $B^4$\\ \end{minipage}}}%
+    \put(1.19167986,1.92661837){\color[rgb]{0,0,0}\makebox(0,0)[lt]{\begin{minipage}{0.79722134\unitlength}\raggedright \end{minipage}}}%
+    \put(0,0){\includegraphics[width=\unitlength,page=2]{link_diagram.pdf}}%
+    \put(0.30826412,1.20312347){\color[rgb]{0,0,0}\makebox(0,0)[lt]{\begin{minipage}{0.77132343\unitlength}\raggedright $B^4$\\ \end{minipage}}}%
+    \put(0,0){\includegraphics[width=\unitlength,page=3]{link_diagram.pdf}}%
+    \put(0.71933102,1.26837147){\color[rgb]{0,0,0}\makebox(0,0)[lt]{\begin{minipage}{0.16148245\unitlength}\raggedright \end{minipage}}}%
+    \put(0.63565374,1.14799364){\color[rgb]{0,0,0}\makebox(0,0)[lt]{\begin{minipage}{0.21139524\unitlength}\raggedright $\Sigma$\end{minipage}}}%
+    \put(0.85145305,1.2507552){\color[rgb]{0,0,0}\makebox(0,0)[lb]{\smash{}}}%
+  \end{picture}%
+\endgroup%

lectures_on_knot_theory.tex

@@ -1,4 +1,5 @@
 \documentclass[12pt, twoside]{article}
+\usepackage[pdf]{pstricks}
 
 \usepackage{amssymb}
 \usepackage{amsmath}
@@ -14,7 +15,8 @@
 \usepackage{advdate}
 \usepackage{amsthm}
 \usepackage[useregional]{datetime2}
-
+\usepackage{tikz}
+\usetikzlibrary{cd}
 \hypersetup{
     colorlinks,
     citecolor=black,
@@ -25,7 +27,6 @@
 \usepackage{fontspec}
 \usepackage{mathtools}
 \usepackage{unicode-math}
-
 \graphicspath{ {images/} }
 
 \newtheorem{lemma}{Lemma}
@@ -47,6 +48,7 @@
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \newcommand{\overbar}[1]{\mkern 1.5mu\overline{\mkern-1.5mu#1\mkern-1.5mu}\mkern 1.5mu}
 \DeclareMathOperator{\Hom}{Hom}
+\DeclareMathOperator{\rank}{rank}
 \AtBeginDocument{\renewcommand{\setminus}{\mathbin{\backslash}}}
 
 \input{knots_macros}
@@ -54,17 +56,15 @@
 \titleformat{\section}{\normalfont \large \bfseries}
 {Lecture\ \thesection}{2.3ex plus .2ex}{} 
 \titlespacing{\subsection}{2em}{*1}{*1}
-
+\usepackage{enumitem}
+\setlist[itemize]{topsep=0pt,before=\leavevmode\vspace{0.5em}}
 
 \begin{document}
-%\tableofcontents
+\tableofcontents
 %\newpage
 %\input{myNotes}
 
-\section{}
-\begin{flushright}
-\DTMdate{2019-02-25}
-\end{flushright}
+\section{\hfill\DTMdate{2019-02-25}}
 \begin{definition}
 A knot $K$ in $S^3$ is a smooth (PL - smooth) embedding of a circle $S^1$ in $S^3$:
 \begin{align*}
@@ -107,17 +107,18 @@ Links:
 \begin{itemize}
 \item
 a trivial link with $3$ components:
-\includegraphics[width=0.13\textwidth]{3unknots.png},
+\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
-Borromean link
+Borromean link:
 \includegraphics[width=0.1\textwidth]{BorromeanRings.png},
 \end{itemize}
 \end{example}
+
 \begin{definition}
 A link diagram is a picture over projection of a link is $S^3$($\mathbb{R}^3$) such that:
 \begin{enumerate}
@@ -132,7 +133,7 @@ ${D_{\pi}}_{\big|L}$ is non degenerate
 \end{definition}
 There are under- and overcrossings  (tunnels and bridges) on a link diagrams with an obvious meaning.\\
 Every link admits a link diagram.
-\begin{comment}
+%\begin{comment}
 
 \subsection{Reidemeister moves}
 A Reidemeister move is one of the three types of operation on a link diagram as shown in Figure~\ref{fig: reidemeister}.
@@ -160,7 +161,7 @@ deformed into each other by a finite sequence of Reidemeister moves (and isotopy
 \end{align*}
 Fact (Milnor Singular Points of Complex Hypersurfaces):
 \end{example}
-\end{comment}
+%\end{comment}
 
 An oriented knot is called negative amphichiral if the mirror image $m(K)$ if $K$ is equivalent the reverse knot of $K$. \\
 \begin{example}[Problem]
@@ -168,10 +169,7 @@ Prove that if $K$ is negative amphichiral, then $K \# K$ in
 $\mathbf{C}$
 \end{example}
 
-\section{}
-\begin{flushright}
-\DTMdate{2019-03-04}
-\end{flushright}
+\section{\hfill\DTMdate{2019-03-04}}
 \begin{proof}("joke")\\
 Let $K \in S^3$ be a knot and $N$ be its tubular neighbourhood. 
 \begin{align*}
@@ -182,10 +180,7 @@ For a pair $(S^3, S^3 \setminus N)$ we have:
 H^0(S^3)
 \end{align*}
 \end{proof}
-\section{}
-\begin{flushright}
-\DTMdate{2019-03-18}
-\end{flushright}
+\section{\hfill\DTMdate{2019-03-18}}
 \begin{definition}
 A knot $K$ is called (smoothly) slice if $K$ is smoothly concordant to an unknot. \\
 A knot $K$ is smoothly slice if and only if $K$ bounds a smoothly embedded disk in $B^4$.
@@ -216,10 +211,7 @@ Are there in concordance group torsion elements that are not $2$ torsion element
 \end{example}
 \noindent
 Remark: $K \sim K^{\prime} \Leftrightarrow K \# -K^{\prime}$ is slice.
-\section{}
-\begin{flushright}
-\DTMdate{2019-04-08}
-\end{flushright}
+\section{\hfill\DTMdate{2019-04-08}}
 $X$ is a closed orientable four-manifold. Assume $\pi_1(X) = 0$ (it is not needed to define the intersection form). In particular $H_1(X) = 0$. 
 $H_2$ is free (exercise). 
 \begin{align*}
@@ -230,10 +222,114 @@ $H_2(X, \mathbb{Z}) \times
 H_2(X, \mathbb{Z}) \longrightarrow \mathbb{Z}$ - symmetric, non singular. 
 \\
 Let $A$ and $B$ be closed, oriented surfaces in $X$. 
-\section{} 
-\begin{flushright}
-\DTMdate{2019-05-20}
-\end{flushright}
+
+
+\section{\hfill\DTMdate{2019-04-15}}
+In other words:\\
+Choose a basis $(b_1, ..., b_i)$ \\
+???\\
+of $H_2(Y, \mathbb{Z}$, then $A = (b_i, b_y)$ \\??\\ is a matrix of intersection form: 
+\begin{align*}
+\quot{\mathbb{Z}^n}{A\mathbb{Z}^n} \cong H_1(Y, \mathbb{Z}).
+\end{align*}
+In particular $\mid \det A\mid = \# H_1(Y, \mathbb{Z}$.\\
+That means - what is happening on boundary is a measure of degeneracy.  
+\\
+\vspace{1cm}
+\begin{center}
+\begin{tikzcd}
+[
+column sep=tiny,
+row sep=small,
+ar symbol/.style = {draw=none,"\textstyle#1" description,sloped},
+isomorphic/.style = {ar symbol={\cong}},
+]
+H_1(Y, \mathbb{Z})&
+\times \quad H_1(Y, \mathbb{Z})&
+\longrightarrow &
+\quot{\mathbb{Q}}{\mathbb{Z}}
+ \text{ - a linking form}
+\\
+\quot{\mathbb{Z}^n}{A\mathbb{Z}} \ar[u,isomorphic] & 
+\quot{\mathbb{Z}^n}{A\mathbb{Z}} \ar[u,isomorphic] &\\
+\end{tikzcd}
+$(a, b) \mapsto aA^{-1}b^T$
+\end{center}
+The intersection form on a four-manifold determines the linking on the boundary. \\
+
+\noindent
+Let $K \in S^1$ be a knot, $\Sigma(K)$ its double branched cover. If $V$ is a Seifert matrix for $K$, then 
+$H_1(\Sigma(K), \mathbb{Z}) \cong  \quot{\mathbb{Z}^n}{A\mathbb{Z}}$ where 
+$A = V \times V^T$, where $n = \rank V$.
+%\input{ink_diag}
+\begin{figure}[h]
+\fontsize{40}{10}\selectfont
+\centering{
+\def\svgwidth{\linewidth}
+\resizebox{0.5\textwidth}{!}{\input{images/ball_4.pdf_tex}}
+\caption{Pushing the Seifert surface in 4-ball.}
+\label{fig:pushSeifert}
+}
+\end{figure}
+\noindent
+Let $X$ be the four-manifold obtained via the double branched cover of $B^4$ branched along $\widetilde{\Sigma}$.
+\begin{fact}
+\begin{itemize}
+\item $X$ is a smooth four-manifold, 
+\item $H_1(X, \mathbb{Z}) =0$, 
+\item $H_2(X, \mathbb{Z}) \cong \mathbb{Z}^n$
+\item The intersection form on $X$ is $V + V^T$.
+\end{itemize}
+\end{fact}
+\noindent
+Let  $Y = \Sigma(K)$. Then:
+\begin{align*}
+&H_1(Y, \mathbb{Z}) \times H_1(Y, \mathbb{Z}) \longrightarrow \quot{\mathbb{Q}}{\mathbb{Z}}\\ &(a,b) \mapsto a A^{-1} b^{T},\qquad
+A = V + V^T\\
+&H_1(Y, \mathbb{Z}) \cong \quot{\mathbb{Z}^n}{A\mathbb{Z}}\\
+&A \longrightarrow BAC^T \quad \text{Smith normal form}
+\end{align*}
+???????????????????????\\
+In general 
+
+\section{\hfill\DTMdate{2019-05-20}}
+
+Let $M$ be closed, oriented, compact four-dimensional manifold.\\
+??????????????????????????????????\\
+If $H_1(M, \mathbb{Z}) = 0$ then there exists a 
+bilinear form - the intersection form on $M$: 
+
+\begin{center}
+\begin{tikzcd}
+[
+column sep=tiny,
+row sep=small,
+ar symbol/.style = {draw=none,"\textstyle#1" description,sloped},
+isomorphic/.style = {ar symbol={\cong}},
+]
+H_2(M, \mathbb{Z})&
+\times & H_2(M, \mathbb{Z})
+\longrightarrow &
+\mathbb{Z}
+\\
+\ar[u,isomorphic] \mathbb{Z}^n  && &\\
+\end{tikzcd}
+\end{center}
+\noindent
+Let us consider a specific case: $M$ has a boundary $Y = \partial M$. 
+\\??????\\
+Betti number $b_1(Y) = 0$, $H_1(Y, \mathbb{Z})$ is  finite. \\
+Then: $H_2(M, \mathbb{Z})
+\times  H_2(M, \mathbb{Z})
+\longrightarrow 
+\mathbb{Z}$ can be degenerate in the sense that
+\begin{align*}
+H_2(M, \mathbb{Z}) \longrightarrow \Hom (H_2(M, \mathbb{Z}), \mathbb{Z})\\
+(a, b) \mapsto \mathbb{Z}\\
+a \mapsto (a, \_) H_2(M, \mathbb{Z})
+\end{align*} has coker 
+
+
 Let $K \subset S^3$  be a knot, \\
 $X = S^3 \setminus K$ - a knot complement,  \\
 $\widetilde{X} \xrightarrow{\enspace \rho \enspace} X$ - an infinite cyclic cover (universal abelian cover).
@@ -249,10 +345,10 @@ H_1(\widetilde{X}, \mathbb{Z}[t, t^{-1}]) \longrightarrow \quot{\mathbb{Q}}{\mat
 
 \begin{fact}
 \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}
+&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*}
-where $V$ is a Seifert matrix.
 \end{fact}
 \begin{fact}
 \begin{align*}
@@ -413,7 +509,7 @@ g\overbar{g}h \equiv 1 \mod{p^{2k}}\\
 g\overbar{g} \equiv 1 \mod{p^k}
 \end{align*}
 ???????????????????????????????\\
-If $p$ has no roots on $S^1$ then $B(z) > 0$ for all $z$, so the assumptions of Lemma \ref{L:coprime polynomials} are satisfied no matter what $A$ is. 
+If $P$ has no roots on $S^1$ then $B(z) > 0$ for all $z$, so the assumptions of Lemma \ref{L:coprime polynomials} are satisfied no matter what $A$ is. 
 \end{proof}
 ?????????????????\\
 \begin{align*}
@@ -448,31 +544,11 @@ $2 \sum\limits_{k_i \text{ odd}} \epsilon_i$. The peak of the signature function
 %$(\eta_{k, \xi_l^{+}} -\eta_{k, \xi_l^{-}}$
 \end{theorem}
 \end{proof}
-\section{}
-\begin{flushright}
-\DTMdate{2019-05-27}
-\end{flushright}
+\section{\hfill\DTMdate{2019-05-27}}
 ....
 \begin{definition}
 A square hermitian matrix $A$ of size $n$.  
 \end{definition}
 
 field of fractions
-
-\section{}
-In other words:\\
-Choose a basis $(b_1, ..., b_i)$ \\
-???\\
-of $H_2(Y, \mathbb{Z}$, then $A = (b_i, b_y)$ \\??\\ is a matrix of intersection form: 
-\begin{align*}
-\quot{\mathbb{Z}^n}{A\mathbb{Z}^n} \cong H_1(Y, \mathbb{Z}).
-\end{align*}
-In particular $\mid \det A\mid = \# H_1(Y, \mathbb{Z}$.\\
-That means - what is happening on boundary is a measure of degeneracy.  
-\\
-\vspace{1cm}
-\begin{align*}
-H_1(Y, \mathbb{Z}) \times
-H_1(Y, \mathbb{Z}) \longrightarrow \quot{\mathbb{Q}}{\mathbb{Z}} \text{ - a linking form}
-\end{align*}
-\end{document}
+\end{document}