some inkspace pictures

This commit is contained in:
Maria Marchwicka 2019-05-29 15:54:14 +02:00
parent b2a61ac843
commit 0583159f09
11 changed files with 313 additions and 54 deletions

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%% 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
%%
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%% 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%
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\providecommand\color[2][]{%
\errmessage{(Inkscape) Color is used for the text in Inkscape, but the package 'color.sty' is not loaded}%
\renewcommand\color[2][]{}%
}%
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\errmessage{(Inkscape) Transparency is used (non-zero) for the text in Inkscape, but the package 'transparent.sty' is not loaded}%
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%% 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%
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\providecommand\color[2][]{%
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\put(0.31120017,0.7509725){\color[rgb]{0,0,0}\makebox(0,0)[lt]{\begin{minipage}{0.77132343\unitlength}\raggedright $B^4$\\ \end{minipage}}}%
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\put(0.30826412,1.20312347){\color[rgb]{0,0,0}\makebox(0,0)[lt]{\begin{minipage}{0.77132343\unitlength}\raggedright $B^4$\\ \end{minipage}}}%
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\put(0.63565374,1.14799364){\color[rgb]{0,0,0}\makebox(0,0)[lt]{\begin{minipage}{0.21139524\unitlength}\raggedright $\Sigma$\end{minipage}}}%
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@ -1,4 +1,5 @@
\documentclass[12pt, twoside]{article} \documentclass[12pt, twoside]{article}
\usepackage[pdf]{pstricks}
\usepackage{amssymb} \usepackage{amssymb}
\usepackage{amsmath} \usepackage{amsmath}
@ -14,7 +15,8 @@
\usepackage{advdate} \usepackage{advdate}
\usepackage{amsthm} \usepackage{amsthm}
\usepackage[useregional]{datetime2} \usepackage[useregional]{datetime2}
\usepackage{tikz}
\usetikzlibrary{cd}
\hypersetup{ \hypersetup{
colorlinks, colorlinks,
citecolor=black, citecolor=black,
@ -25,7 +27,6 @@
\usepackage{fontspec} \usepackage{fontspec}
\usepackage{mathtools} \usepackage{mathtools}
\usepackage{unicode-math} \usepackage{unicode-math}
\graphicspath{ {images/} } \graphicspath{ {images/} }
\newtheorem{lemma}{Lemma} \newtheorem{lemma}{Lemma}
@ -47,6 +48,7 @@
%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%
\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}
\DeclareMathOperator{\Hom}{Hom} \DeclareMathOperator{\Hom}{Hom}
\DeclareMathOperator{\rank}{rank}
\AtBeginDocument{\renewcommand{\setminus}{\mathbin{\backslash}}} \AtBeginDocument{\renewcommand{\setminus}{\mathbin{\backslash}}}
\input{knots_macros} \input{knots_macros}
@ -54,17 +56,15 @@
\titleformat{\section}{\normalfont \large \bfseries} \titleformat{\section}{\normalfont \large \bfseries}
{Lecture\ \thesection}{2.3ex plus .2ex}{} {Lecture\ \thesection}{2.3ex plus .2ex}{}
\titlespacing{\subsection}{2em}{*1}{*1} \titlespacing{\subsection}{2em}{*1}{*1}
\usepackage{enumitem}
\setlist[itemize]{topsep=0pt,before=\leavevmode\vspace{0.5em}}
\begin{document} \begin{document}
%\tableofcontents \tableofcontents
%\newpage %\newpage
%\input{myNotes} %\input{myNotes}
\section{} \section{\hfill\DTMdate{2019-02-25}}
\begin{flushright}
\DTMdate{2019-02-25}
\end{flushright}
\begin{definition} \begin{definition}
A knot $K$ in $S^3$ is a smooth (PL - smooth) embedding of a circle $S^1$ in $S^3$: A knot $K$ in $S^3$ is a smooth (PL - smooth) embedding of a circle $S^1$ in $S^3$:
\begin{align*} \begin{align*}
@ -107,17 +107,18 @@ Links:
\begin{itemize} \begin{itemize}
\item \item
a trivial link with $3$ components: a trivial link with $3$ components:
\includegraphics[width=0.13\textwidth]{3unknots.png}, \includegraphics[width=0.2\textwidth]{3unknots.png},
\item \item
a hopf link: \includegraphics[width=0.13\textwidth]{Hopf.png}, a hopf link: \includegraphics[width=0.13\textwidth]{Hopf.png},
\item \item
a Whitehead link: a Whitehead link:
\includegraphics[width=0.13\textwidth]{WhiteheadLink.png}, \includegraphics[width=0.13\textwidth]{WhiteheadLink.png},
\item \item
Borromean link 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}
@ -132,7 +133,7 @@ ${D_{\pi}}_{\big|L}$ is non degenerate
\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} \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}. 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*} \end{align*}
Fact (Milnor Singular Points of Complex Hypersurfaces): Fact (Milnor Singular Points of Complex Hypersurfaces):
\end{example} \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 $K$. \\ 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] \begin{example}[Problem]
@ -168,10 +169,7 @@ Prove that if $K$ is negative amphichiral, then $K \# K$ in
$\mathbf{C}$ $\mathbf{C}$
\end{example} \end{example}
\section{} \section{\hfill\DTMdate{2019-03-04}}
\begin{flushright}
\DTMdate{2019-03-04}
\end{flushright}
\begin{proof}("joke")\\ \begin{proof}("joke")\\
Let $K \in S^3$ be a knot and $N$ be its tubular neighbourhood. Let $K \in S^3$ be a knot and $N$ be its tubular neighbourhood.
\begin{align*} \begin{align*}
@ -182,10 +180,7 @@ For a pair $(S^3, S^3 \setminus N)$ we have:
H^0(S^3) H^0(S^3)
\end{align*} \end{align*}
\end{proof} \end{proof}
\section{} \section{\hfill\DTMdate{2019-03-18}}
\begin{flushright}
\DTMdate{2019-03-18}
\end{flushright}
\begin{definition} \begin{definition}
A knot $K$ is called (smoothly) slice if $K$ is smoothly concordant to an unknot. \\ 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 $B^4$. 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} \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{} \section{\hfill\DTMdate{2019-04-08}}
\begin{flushright}
\DTMdate{2019-04-08}
\end{flushright}
$X$ 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*}
@ -230,10 +222,114 @@ $H_2(X, \mathbb{Z}) \times
H_2(X, \mathbb{Z}) \longrightarrow \mathbb{Z}$ - symmetric, non singular. H_2(X, \mathbb{Z}) \longrightarrow \mathbb{Z}$ - symmetric, non singular.
\\ \\
Let $A$ and $B$ be closed, oriented surfaces in $X$. Let $A$ and $B$ be closed, oriented surfaces in $X$.
\section{}
\begin{flushright}
\DTMdate{2019-05-20} \section{\hfill\DTMdate{2019-04-15}}
\end{flushright} 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, \\ 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).
@ -249,10 +345,10 @@ H_1(\widetilde{X}, \mathbb{Z}[t, t^{-1}]) \longrightarrow \quot{\mathbb{Q}}{\mat
\begin{fact} \begin{fact}
\begin{align*} \begin{align*}
H_1(\widetilde{X}, \mathbb{Z}[t, t^{-1}]) \cong &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} \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*} \end{align*}
where $V$ is a Seifert matrix.
\end{fact} \end{fact}
\begin{fact} \begin{fact}
\begin{align*} \begin{align*}
@ -413,7 +509,7 @@ g\overbar{g}h \equiv 1 \mod{p^{2k}}\\
g\overbar{g} \equiv 1 \mod{p^k} g\overbar{g} \equiv 1 \mod{p^k}
\end{align*} \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} \end{proof}
?????????????????\\ ?????????????????\\
\begin{align*} \begin{align*}
@ -448,31 +544,11 @@ $2 \sum\limits_{k_i \text{ odd}} \epsilon_i$. The peak of the signature function
%$(\eta_{k, \xi_l^{+}} -\eta_{k, \xi_l^{-}}$ %$(\eta_{k, \xi_l^{+}} -\eta_{k, \xi_l^{-}}$
\end{theorem} \end{theorem}
\end{proof} \end{proof}
\section{} \section{\hfill\DTMdate{2019-05-27}}
\begin{flushright}
\DTMdate{2019-05-27}
\end{flushright}
.... ....
\begin{definition} \begin{definition}
A square hermitian matrix $A$ of size $n$. A square hermitian matrix $A$ of size $n$.
\end{definition} \end{definition}
field of fractions field of fractions
\end{document}
\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}