some inkspace pictures
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Maria Marchwicka
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0583159f09
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@ -1,22 +1,39 @@
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\documentclass[12pt, twoside]{article}
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\usepackage[pdf]{pstricks}
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\usepackage{amssymb}
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\usepackage{amsmath}
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\usepackage{xfrac}
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\usepackage[english]{babel}
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\usepackage{csquotes}
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\usepackage{graphicx}
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\usepackage{float}
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\usepackage{titlesec}
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\usepackage{comment}
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\usepackage{pict2e}
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\usepackage{hyperref}
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\usepackage{advdate}
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\usepackage{amsthm}
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\usepackage[english]{babel}
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\usepackage{comment}
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\usepackage{csquotes}
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\usepackage[useregional]{datetime2}
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\usepackage{enumitem}
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\usepackage{fontspec}
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\usepackage{float}
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\usepackage{graphicx}
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\usepackage{hyperref}
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\usepackage{mathtools}
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\usepackage{pict2e}
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\usepackage[pdf]{pstricks}
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\usepackage{tikz}
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\usepackage{titlesec}
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\usepackage{xfrac}
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\usepackage{unicode-math}
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\usetikzlibrary{cd}
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\hypersetup{
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colorlinks,
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citecolor=black,
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@ -24,40 +41,45 @@
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linkcolor=black,
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urlcolor=black
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}
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\usepackage{fontspec}
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\usepackage{mathtools}
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\usepackage{unicode-math}
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\graphicspath{ {images/} }
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\newtheoremstyle{break}
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{\topsep}{\topsep}%
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{\itshape}{}%
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{\bfseries}{}%
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{\newline}{}%
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\theoremstyle{break}
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\newtheorem{lemma}{Lemma}
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\newtheorem{fact}{Fact}
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\newtheorem{example}{Example}
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%\theoremstyle{definition}
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\newtheorem{definition}{Definition}
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%\theoremstyle{plain}
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\newtheorem{theorem}{Theorem}
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\newtheorem{proposition}{Proposition}
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\newcommand{\contradiction}{%
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\ensuremath{{\Rightarrow\mspace{-2mu}\Leftarrow}}%
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}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%%%% For quotient groups / modding equiv relations
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%%%% Use: \quot{A}{B} --> A/B
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\newcommand{\contradiction}{%
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\ensuremath{{\Rightarrow\mspace{-2mu}\Leftarrow}}}
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\newcommand*\quot[2]{{^{\textstyle #1}\big/_{\textstyle #2}}}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\newcommand{\overbar}[1]{\mkern 1.5mu\overline{\mkern-1.5mu#1\mkern-1.5mu}\mkern 1.5mu}
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\newcommand{\overbar}[1]{%
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\mkern 1.5mu=\overline{%
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\mkern-1.5mu#1\mkern-1.5mu}%
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\mkern 1.5mu}
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\AtBeginDocument{\renewcommand{\setminus}{%
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\mathbin{\backslash}}}
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\DeclareMathOperator{\Hom}{Hom}
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\DeclareMathOperator{\rank}{rank}
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\AtBeginDocument{\renewcommand{\setminus}{\mathbin{\backslash}}}
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\titleformat{\section}{\normalfont \large \bfseries}{%
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Lecture\ \thesection}{2.3ex plus .2ex}{}
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%\setlist[itemize]{topsep=0pt,before=%\leavevmode\vspace{0.5em}}
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\input{knots_macros}
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\graphicspath{ {images/} }
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\titleformat{\section}{\normalfont \large \bfseries}
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{Lecture\ \thesection}{2.3ex plus .2ex}{}
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\titlespacing{\subsection}{2em}{*1}{*1}
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\usepackage{enumitem}
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\setlist[itemize]{topsep=0pt,before=\leavevmode\vspace{0.5em}}
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\begin{document}
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\tableofcontents
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@ -74,8 +96,18 @@ A knot $K$ in $S^3$ is a smooth (PL - smooth) embedding of a circle $S^1$ in $S^
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\noindent
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Usually we think about a knot as an image of an embedding: $K = \varphi(S^1)$.
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\begin{example}
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!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
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knot and not a knot (not inection), not smooth,
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\begin{itemize}
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\item
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Knots:
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\includegraphics[width=0.08\textwidth]{unknot.png},
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\includegraphics[width=0.08\textwidth]{trefoil.png}.
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\item
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Not knots:
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\includegraphics[width=0.12\textwidth]{not_injective_knot.png}
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(it is not an injection),
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\includegraphics[width=0.08\textwidth]{not_smooth_knot.png}
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(it is not smooth).
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\end{itemize}
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\end{example}
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\begin{definition}
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%\hfill\\
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@ -89,9 +121,10 @@ $\Phi_1 = \varphi_1$.
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\end{definition}
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\begin{theorem}
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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:
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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:
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\begin{align*}
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&\Psi: S^3 \hookrightarrow S^3,\\
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&\psi(t) = \psi_t \text{ is continius on $t\in [0,1]$}\\
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&\psi_t: S^3 \hookrightarrow S^3,\\
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& \psi_0 = id ,\\
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& \psi_1(K_0) = K_1.
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\end{align*}
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@ -100,7 +133,7 @@ Two knots $K_0$ and $K_1$ are isotopic if and only if they are ambient isotopic,
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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 $S^1$.
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\end{definition}
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\begin{definition}
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A link with k - components is a (smooth) embedding of\\ $\overbrace{S^1 \sqcup \ldots \sqcup S^1}^k$ in $S^3$
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A link with k - components is a (smooth) embedding of $\overbrace{S^1 \sqcup \ldots \sqcup S^1}^k$ in $S^3$
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\end{definition}
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\begin{example}
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Links:
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@ -118,24 +151,23 @@ Borromean link:
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\includegraphics[width=0.1\textwidth]{BorromeanRings.png},
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\end{itemize}
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\end{example}
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%
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%
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%
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\begin{definition}
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A link diagram is a picture over projection of a link is $S^3$($\mathbb{R}^3$) such that:
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\begin{enumerate}
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\begin{enumerate}[label={(\arabic*)}]
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\item
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${D_{\pi}}_{\big|L}$ is non degenerate
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\includegraphics[width=0.02\textwidth]{LinkDiagram1.png},
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\item the double points are not degenerate
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\includegraphics[width=0.02\textwidth]{LinkDiagram2.png},
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\item there are no triple point
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\includegraphics[width=0.03\textwidth]{LinkDiagram3.png}.
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${D_{\pi}}_{\big|L}$ is non degenerate: \includegraphics[width=0.05\textwidth]{LinkDiagram1.png},
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\item the double points are not degenerate: \includegraphics[width=0.05\textwidth]{LinkDiagram2.png},
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\item there are no triple point: \includegraphics[width=0.05\textwidth]{LinkDiagram3.png}.
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\end{enumerate}
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\end{definition}
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There are under- and overcrossings (tunnels and bridges) on a link diagrams with an obvious meaning.\\
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Every link admits a link diagram.
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%\begin{comment}
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\subsection{Reidemeister moves}
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\section*{Reidemeister moves}
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A Reidemeister move is one of the three types of operation on a link diagram as shown in Figure~\ref{fig: reidemeister}.
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%
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The first Reidemeister move inserts or removes a coil.
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@ -211,7 +243,12 @@ Are there in concordance group torsion elements that are not $2$ torsion element
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\end{example}
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\noindent
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Remark: $K \sim K^{\prime} \Leftrightarrow K \# -K^{\prime}$ is slice.
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%
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%
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\section{\hfill\DTMdate{2019-04-08}}
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%
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%
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$X$ is a closed orientable four-manifold. Assume $\pi_1(X) = 0$ (it is not needed to define the intersection form). In particular $H_1(X) = 0$.
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$H_2$ is free (exercise).
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\begin{align*}
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@ -262,7 +299,7 @@ Let $K \in S^1$ be a knot, $\Sigma(K)$ its double branched cover. If $V$ is a Se
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$H_1(\Sigma(K), \mathbb{Z}) \cong \quot{\mathbb{Z}^n}{A\mathbb{Z}}$ where
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$A = V \times V^T$, where $n = \rank V$.
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%\input{ink_diag}
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\begin{figure}[h]
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\begin{figure}[H]
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\fontsize{40}{10}\selectfont
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\centering{
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\def\svgwidth{\linewidth}
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@ -294,7 +331,7 @@ In general
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\section{\hfill\DTMdate{2019-05-20}}
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Let $M$ be closed, oriented, compact four-dimensional manifold.\\
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Let $M$ be compact, oriented, connected four-dimensional manifold.\\
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??????????????????????????????????\\
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If $H_1(M, \mathbb{Z}) = 0$ then there exists a
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bilinear form - the intersection form on $M$:
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@ -327,9 +364,9 @@ Then: $H_2(M, \mathbb{Z})
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H_2(M, \mathbb{Z}) \longrightarrow \Hom (H_2(M, \mathbb{Z}), \mathbb{Z})\\
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(a, b) \mapsto \mathbb{Z}\\
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a \mapsto (a, \_) H_2(M, \mathbb{Z})
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\end{align*} has coker
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\end{align*}
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has coker precisely $H_1(Y, \mathbb{Z})$.
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\\???????????????\\
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Let $K \subset S^3$ be a knot, \\
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$X = S^3 \setminus K$ - a knot complement, \\
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$\widetilde{X} \xrightarrow{\enspace \rho \enspace} X$ - an infinite cyclic cover (universal abelian cover).
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@ -551,4 +588,8 @@ A square hermitian matrix $A$ of size $n$.
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\end{definition}
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field of fractions
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\section{balagan}
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\end{document}
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