diff --git a/images/3unknots.png b/images/3unknots.png index aaf791c..5a88cd8 100644 Binary files a/images/3unknots.png and b/images/3unknots.png differ diff --git a/images/BorromeanRings.png b/images/BorromeanRings.png index 36e252a..b7e0bbf 100644 Binary files a/images/BorromeanRings.png and b/images/BorromeanRings.png differ diff --git a/images/Hopf.png b/images/Hopf.png index 96d429f..7b6971e 100644 Binary files a/images/Hopf.png and b/images/Hopf.png differ diff --git a/images/WhiteheadLink.png b/images/WhiteheadLink.png index eecf317..646e877 100644 Binary files a/images/WhiteheadLink.png and b/images/WhiteheadLink.png differ diff --git a/images/ball_4.pdf b/images/ball_4.pdf new file mode 100644 index 0000000..c8ab754 Binary files /dev/null and b/images/ball_4.pdf differ diff --git a/images/ball_4.pdf_tex b/images/ball_4.pdf_tex new file mode 100644 index 0000000..bd0f8ef --- /dev/null +++ b/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{.pdf_tex} +%% instead of +%% \includegraphics{.pdf} +%% To scale the image, write +%% \def\svgwidth{} +%% \input{.pdf_tex} +%% instead of +%% \includegraphics[width=]{.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{}{.pdf_tex} +%% Alternatively, one can specify +%% \graphicspath{{/}} +%% +%% 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}}}% + 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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}}% + 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image file 'link_diagram.pdf' (pdf, eps, ps) +%% +%% To include the image in your LaTeX document, write +%% \input{.pdf_tex} +%% instead of +%% \includegraphics{.pdf} +%% To scale the image, write +%% \def\svgwidth{} +%% \input{.pdf_tex} +%% instead of +%% \includegraphics[width=]{.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{}{.pdf_tex} +%% Alternatively, one can specify +%% \graphicspath{{/}} +%% +%% 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% diff --git a/lectures_on_knot_theory.tex b/lectures_on_knot_theory.tex index c9d37b2..ca1309a 100644 --- a/lectures_on_knot_theory.tex +++ b/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} \ No newline at end of file +\end{document}