diff --git a/lectures_on_knot_theory.tex b/lectures_on_knot_theory.tex index ca1309a..10e44b2 100644 --- a/lectures_on_knot_theory.tex +++ b/lectures_on_knot_theory.tex @@ -1,22 +1,39 @@ \documentclass[12pt, twoside]{article} -\usepackage[pdf]{pstricks} + \usepackage{amssymb} \usepackage{amsmath} -\usepackage{xfrac} -\usepackage[english]{babel} -\usepackage{csquotes} -\usepackage{graphicx} -\usepackage{float} -\usepackage{titlesec} -\usepackage{comment} -\usepackage{pict2e} -\usepackage{hyperref} \usepackage{advdate} \usepackage{amsthm} + +\usepackage[english]{babel} + +\usepackage{comment} +\usepackage{csquotes} + \usepackage[useregional]{datetime2} + +\usepackage{enumitem} +\usepackage{fontspec} +\usepackage{float} + +\usepackage{graphicx} +\usepackage{hyperref} + +\usepackage{mathtools} + +\usepackage{pict2e} +\usepackage[pdf]{pstricks} + \usepackage{tikz} +\usepackage{titlesec} + +\usepackage{xfrac} + +\usepackage{unicode-math} + \usetikzlibrary{cd} + \hypersetup{ colorlinks, citecolor=black, @@ -24,40 +41,45 @@ linkcolor=black, urlcolor=black } -\usepackage{fontspec} -\usepackage{mathtools} -\usepackage{unicode-math} -\graphicspath{ {images/} } +\newtheoremstyle{break} + {\topsep}{\topsep}% + {\itshape}{}% + {\bfseries}{}% + {\newline}{}% +\theoremstyle{break} \newtheorem{lemma}{Lemma} \newtheorem{fact}{Fact} \newtheorem{example}{Example} -%\theoremstyle{definition} \newtheorem{definition}{Definition} -%\theoremstyle{plain} \newtheorem{theorem}{Theorem} \newtheorem{proposition}{Proposition} -\newcommand{\contradiction}{% - \ensuremath{{\Rightarrow\mspace{-2mu}\Leftarrow}}% -} -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -%%%% For quotient groups / modding equiv relations -%%%% Use: \quot{A}{B} --> A/B +\newcommand{\contradiction}{% + \ensuremath{{\Rightarrow\mspace{-2mu}\Leftarrow}}} \newcommand*\quot[2]{{^{\textstyle #1}\big/_{\textstyle #2}}} -%%%%%%%%%%%%%%%%%%%%%%%%%%%% -\newcommand{\overbar}[1]{\mkern 1.5mu\overline{\mkern-1.5mu#1\mkern-1.5mu}\mkern 1.5mu} +\newcommand{\overbar}[1]{% + \mkern 1.5mu=\overline{% + \mkern-1.5mu#1\mkern-1.5mu}% + \mkern 1.5mu} + +\AtBeginDocument{\renewcommand{\setminus}{% + \mathbin{\backslash}}} + + \DeclareMathOperator{\Hom}{Hom} \DeclareMathOperator{\rank}{rank} -\AtBeginDocument{\renewcommand{\setminus}{\mathbin{\backslash}}} + + +\titleformat{\section}{\normalfont \large \bfseries}{% + Lecture\ \thesection}{2.3ex plus .2ex}{} + +%\setlist[itemize]{topsep=0pt,before=%\leavevmode\vspace{0.5em}} + \input{knots_macros} +\graphicspath{ {images/} } -\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 @@ -74,8 +96,18 @@ A knot $K$ in $S^3$ is a smooth (PL - smooth) embedding of a circle $S^1$ in $S^ \noindent Usually we think about a knot as an image of an embedding: $K = \varphi(S^1)$. \begin{example} -!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! -knot and not a knot (not inection), not smooth, +\begin{itemize} +\item +Knots: +\includegraphics[width=0.08\textwidth]{unknot.png}, +\includegraphics[width=0.08\textwidth]{trefoil.png}. +\item +Not knots: +\includegraphics[width=0.12\textwidth]{not_injective_knot.png} +(it is not an injection), +\includegraphics[width=0.08\textwidth]{not_smooth_knot.png} +(it is not smooth). +\end{itemize} \end{example} \begin{definition} %\hfill\\ @@ -89,9 +121,10 @@ $\Phi_1 = \varphi_1$. \end{definition} \begin{theorem} -Two knots $K_0$ and $K_1$ are isotopic if and only if they are ambient isotopic, i.e. there exists a family of self-diffeomorphisms $\Psi$ such that: +Two knots $K_0$ and $K_1$ are isotopic if and only if they are ambient isotopic, i.e. there exists a family of self-diffeomorphisms $\Psi = \{\psi_t: t \in [0, 1]\}$ such that: \begin{align*} -&\Psi: S^3 \hookrightarrow S^3,\\ +&\psi(t) = \psi_t \text{ is continius on $t\in [0,1]$}\\ +&\psi_t: S^3 \hookrightarrow S^3,\\ & \psi_0 = id ,\\ & \psi_1(K_0) = K_1. \end{align*} @@ -100,7 +133,7 @@ Two knots $K_0$ and $K_1$ are isotopic if and only if they are ambient isotopic, A knot is trivial (unknot) if it is equivalent to an embedding $\varphi(t) = (\cos t, \sin t, 0)$, where $t \in [0, 2 \pi] $ is a parametrisation of $S^1$. \end{definition} \begin{definition} -A link with k - components is a (smooth) embedding of\\ $\overbrace{S^1 \sqcup \ldots \sqcup S^1}^k$ in $S^3$ +A link with k - components is a (smooth) embedding of $\overbrace{S^1 \sqcup \ldots \sqcup S^1}^k$ in $S^3$ \end{definition} \begin{example} Links: @@ -118,24 +151,23 @@ 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} +\begin{enumerate}[label={(\arabic*)}] \item -${D_{\pi}}_{\big|L}$ is non degenerate -\includegraphics[width=0.02\textwidth]{LinkDiagram1.png}, -\item the double points are not degenerate -\includegraphics[width=0.02\textwidth]{LinkDiagram2.png}, -\item there are no triple point -\includegraphics[width=0.03\textwidth]{LinkDiagram3.png}. +${D_{\pi}}_{\big|L}$ is non degenerate: \includegraphics[width=0.05\textwidth]{LinkDiagram1.png}, +\item the double points are not degenerate: \includegraphics[width=0.05\textwidth]{LinkDiagram2.png}, +\item there are no triple point: \includegraphics[width=0.05\textwidth]{LinkDiagram3.png}. \end{enumerate} \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} -\subsection{Reidemeister moves} +\section*{Reidemeister moves} A Reidemeister move is one of the three types of operation on a link diagram as shown in Figure~\ref{fig: reidemeister}. % The first Reidemeister move inserts or removes a coil. @@ -211,7 +243,12 @@ 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{\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*} @@ -262,7 +299,7 @@ Let $K \in S^1$ be a knot, $\Sigma(K)$ its double branched cover. If $V$ is a Se $H_1(\Sigma(K), \mathbb{Z}) \cong \quot{\mathbb{Z}^n}{A\mathbb{Z}}$ where $A = V \times V^T$, where $n = \rank V$. %\input{ink_diag} -\begin{figure}[h] +\begin{figure}[H] \fontsize{40}{10}\selectfont \centering{ \def\svgwidth{\linewidth} @@ -294,7 +331,7 @@ In general \section{\hfill\DTMdate{2019-05-20}} -Let $M$ be closed, oriented, compact four-dimensional manifold.\\ +Let $M$ be compact, oriented, connected four-dimensional manifold.\\ ??????????????????????????????????\\ If $H_1(M, \mathbb{Z}) = 0$ then there exists a bilinear form - the intersection form on $M$: @@ -327,9 +364,9 @@ Then: $H_2(M, \mathbb{Z}) 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 - - +\end{align*} +has coker precisely $H_1(Y, \mathbb{Z})$. +\\???????????????\\ 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). @@ -551,4 +588,8 @@ A square hermitian matrix $A$ of size $n$. \end{definition} field of fractions + + +\section{balagan} + \end{document}