marchwicka/lectures_on_knot_theory
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some inkspace pictures

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Maria Marchwicka 2019-05-29 20:16:36 +02:00
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lectures_on_knot_theory.tex
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@ -1,22 +1,39 @@
\documentclass[12pt, twoside]{article} \documentclass[12pt, twoside]{article}
\usepackage[pdf]{pstricks}
\usepackage{amssymb} \usepackage{amssymb}
\usepackage{amsmath} \usepackage{amsmath}
\usepackage{xfrac}
\usepackage[english]{babel}
\usepackage{csquotes}
\usepackage{graphicx}
\usepackage{float}
\usepackage{titlesec}
\usepackage{comment}
\usepackage{pict2e}
\usepackage{hyperref}
\usepackage{advdate} \usepackage{advdate}
\usepackage{amsthm} \usepackage{amsthm}
\usepackage[english]{babel}
\usepackage{comment}
\usepackage{csquotes}
\usepackage[useregional]{datetime2} \usepackage[useregional]{datetime2}
\usepackage{enumitem}
\usepackage{fontspec}
\usepackage{float}
\usepackage{graphicx}
\usepackage{hyperref}
\usepackage{mathtools}
\usepackage{pict2e}
\usepackage[pdf]{pstricks}
\usepackage{tikz} \usepackage{tikz}
\usepackage{titlesec}
\usepackage{xfrac}
\usepackage{unicode-math}
\usetikzlibrary{cd} \usetikzlibrary{cd}
\hypersetup{ \hypersetup{
colorlinks, colorlinks,
citecolor=black, citecolor=black,
@ -24,40 +41,45 @@
linkcolor=black, linkcolor=black,
urlcolor=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{lemma}{Lemma}
\newtheorem{fact}{Fact} \newtheorem{fact}{Fact}
\newtheorem{example}{Example} \newtheorem{example}{Example}
%\theoremstyle{definition}
\newtheorem{definition}{Definition} \newtheorem{definition}{Definition}
%\theoremstyle{plain}
\newtheorem{theorem}{Theorem} \newtheorem{theorem}{Theorem}
\newtheorem{proposition}{Proposition} \newtheorem{proposition}{Proposition}
\newcommand{\contradiction}{%
\ensuremath{{\Rightarrow\mspace{-2mu}\Leftarrow}}%
}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \newcommand{\contradiction}{%
%%%% For quotient groups / modding equiv relations \ensuremath{{\Rightarrow\mspace{-2mu}\Leftarrow}}}
%%%% Use: \quot{A}{B} --> A/B
\newcommand*\quot[2]{{^{\textstyle #1}\big/_{\textstyle #2}}} \newcommand*\quot[2]{{^{\textstyle #1}\big/_{\textstyle #2}}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%% \newcommand{\overbar}[1]{%
\newcommand{\overbar}[1]{\mkern 1.5mu\overline{\mkern-1.5mu#1\mkern-1.5mu}\mkern 1.5mu} \mkern 1.5mu=\overline{%
\mkern-1.5mu#1\mkern-1.5mu}%
\mkern 1.5mu}
\AtBeginDocument{\renewcommand{\setminus}{%
\mathbin{\backslash}}}
\DeclareMathOperator{\Hom}{Hom} \DeclareMathOperator{\Hom}{Hom}
\DeclareMathOperator{\rank}{rank} \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} \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} \begin{document}
\tableofcontents \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 \noindent
Usually we think about a knot as an image of an embedding: $K = \varphi(S^1)$. Usually we think about a knot as an image of an embedding: $K = \varphi(S^1)$.
\begin{example} \begin{example}
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! \begin{itemize}
knot and not a knot (not inection), not smooth, \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} \end{example}
\begin{definition} \begin{definition}
%\hfill\\ %\hfill\\
@ -89,9 +121,10 @@ $\Phi_1 = \varphi_1$.
\end{definition} \end{definition}
\begin{theorem} \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*} \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_0 = id ,\\
& \psi_1(K_0) = K_1. & \psi_1(K_0) = K_1.
\end{align*} \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 $S^1$. 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} \end{definition}
\begin{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} \end{definition}
\begin{example} \begin{example}
Links: Links:
@ -118,24 +151,23 @@ Borromean link:
\includegraphics[width=0.1\textwidth]{BorromeanRings.png}, \includegraphics[width=0.1\textwidth]{BorromeanRings.png},
\end{itemize} \end{itemize}
\end{example} \end{example}
%
%
%
\begin{definition} \begin{definition}
A link diagram is a picture over projection of a link is $S^3$($\mathbb{R}^3$) such that: 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 \item
${D_{\pi}}_{\big|L}$ is non degenerate ${D_{\pi}}_{\big|L}$ is non degenerate: \includegraphics[width=0.05\textwidth]{LinkDiagram1.png},
\includegraphics[width=0.02\textwidth]{LinkDiagram1.png}, \item the double points are not degenerate: \includegraphics[width=0.05\textwidth]{LinkDiagram2.png},
\item the double points are not degenerate \item there are no triple point: \includegraphics[width=0.05\textwidth]{LinkDiagram3.png}.
\includegraphics[width=0.02\textwidth]{LinkDiagram2.png},
\item there are no triple point
\includegraphics[width=0.03\textwidth]{LinkDiagram3.png}.
\end{enumerate} \end{enumerate}
\end{definition} \end{definition}
There are under- and overcrossings (tunnels and bridges) on a link diagrams with an obvious meaning.\\ There are under- and overcrossings (tunnels and bridges) on a link diagrams with an obvious meaning.\\
Every link admits a link diagram. Every link admits a link diagram.
%\begin{comment} %\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}. 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. 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} \end{example}
\noindent \noindent
Remark: $K \sim K^{\prime} \Leftrightarrow K \# -K^{\prime}$ is slice. Remark: $K \sim K^{\prime} \Leftrightarrow K \# -K^{\prime}$ is slice.
%
%
\section{\hfill\DTMdate{2019-04-08}} \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 $H_1(X) = 0$. $X$ 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). $H_2$ is free (exercise).
\begin{align*} \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 $H_1(\Sigma(K), \mathbb{Z}) \cong \quot{\mathbb{Z}^n}{A\mathbb{Z}}$ where
$A = V \times V^T$, where $n = \rank V$. $A = V \times V^T$, where $n = \rank V$.
%\input{ink_diag} %\input{ink_diag}
\begin{figure}[h] \begin{figure}[H]
\fontsize{40}{10}\selectfont \fontsize{40}{10}\selectfont
\centering{ \centering{
\def\svgwidth{\linewidth} \def\svgwidth{\linewidth}
@ -294,7 +331,7 @@ In general
\section{\hfill\DTMdate{2019-05-20}} \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 If $H_1(M, \mathbb{Z}) = 0$ then there exists a
bilinear form - the intersection form on $M$: 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})\\ H_2(M, \mathbb{Z}) \longrightarrow \Hom (H_2(M, \mathbb{Z}), \mathbb{Z})\\
(a, b) \mapsto \mathbb{Z}\\ (a, b) \mapsto \mathbb{Z}\\
a \mapsto (a, \_) H_2(M, \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, \\ Let $K \subset S^3$ be a knot, \\
$X = S^3 \setminus K$ - a knot complement, \\ $X = S^3 \setminus K$ - a knot complement, \\
$\widetilde{X} \xrightarrow{\enspace \rho \enspace} X$ - an infinite cyclic cover (universal abelian cover). $\widetilde{X} \xrightarrow{\enspace \rho \enspace} X$ - an infinite cyclic cover (universal abelian cover).
@ -551,4 +588,8 @@ A square hermitian matrix $A$ of size $n$.
\end{definition} \end{definition}
field of fractions field of fractions
\section{balagan}
\end{document} \end{document}