722 lines
23 KiB
TeX
722 lines
23 KiB
TeX
\documentclass[12pt, twoside]{article}
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\usepackage{amssymb}
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\usepackage[useregional]{datetime2}
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\usepackage{graphicx}
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\usepackage{pict2e}
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\usepackage[section]{placeins}
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\usepackage[pdf]{pstricks}
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\usepackage{tikz}
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\usepackage{titlesec}
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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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filecolor=black,
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linkcolor=black,
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urlcolor=black
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\newtheoremstyle{break}
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{\topsep}{\topsep}%
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{\newline}{}%
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\theoremstyle{break}
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\newtheorem{lemma}{Lemma}[section]
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\newtheorem{fact}{Fact}[section]
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\newtheorem{corollary}{Corollary}[section]
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\newtheorem{proposition}{Proposition}[section]
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\newtheorem{example}{Example}[section]
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\newtheorem{problem}{Problem}[section]
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\newtheorem{definition}{Definition}[section]
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\newtheorem{theorem}{Theorem}[section]
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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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\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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\newcommand{\sdots}{\smash{\vdots}}
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\DeclareRobustCommand\longtwoheadrightarrow
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{\relbar\joinrel\twoheadrightarrow}
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\newcommand{\longhookrightarrow}{\lhook\joinrel\longrightarrow}
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\newcommand{\longhookleftarrow}{\longleftarrow\joinrel\rhook}
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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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\DeclareMathOperator{\coker}{coker}
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\DeclareMathOperator{\ord}{ord}
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\DeclareMathOperator{\mytop}{top}
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\DeclareMathOperator{\Gl}{GL}
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\DeclareMathOperator{\Sl}{SL}
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\DeclareMathOperator{\Lk}{lk}
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\DeclareMathOperator{\pt}{\{pt\}}
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\DeclareMathOperator{\sign}{sign}
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\titleformat{\subsection}{%
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\normalfont \fontsize{12}{15}\bfseries}{%
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}{.0ex plus .2ex}{}
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\titleformat{\section}{%
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\normalfont \fontsize{13}{15} \bfseries}{%
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Lecture\ \thesection}%
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{2.3ex plus .2ex}{}
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\titlespacing*{\section}
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{0pt}{16.5ex plus 1ex minus .2ex}{4.3ex plus .2ex}
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\setlist[itemize]{topsep=0pt,before=%
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\leavevmode\vspace{0.5em}}
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\input{knots_macros}
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\graphicspath{ {images/} }
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\begin{document}
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\tableofcontents
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%\newpage
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%\input{myNotes}
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\section{Basic definitions \hfill\DTMdate{2019-02-25}}
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\input{lec_1.tex}
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\section{Alexander polynomial \hfill\DTMdate{2019-03-04}}
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\input{lec_2.tex}
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%add Hurewicz theorem?
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\section{Examples of knot classes
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\hfill\DTMdate{2019-03-11}}
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\input{lec_3.tex}
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\section{Concordance group \hfill\DTMdate{2019-03-18}}
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\input{lec_4.tex}
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\section{Genus $g$ cobordism \hfill\DTMdate{2019-03-25}}
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\input{lec_5.tex}
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\section{\hfill\DTMdate{2019-04-08}}
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\input{lec_6.tex}
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\section{\hfill\DTMdate{2019-04-15}}
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\begin{theorem}
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Suppose that $K \subset S^3$ is a slice knot (i.e. $K$ bound a disk in $B^4$).
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Then if $F$ is a Seifert surface of $K$ and $V$ denotes the associated Seifet matrix, then there exists $P \in \Gl_g(\mathbb{Z})$ such that:
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\\??????????????? T ????????
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\begin{align}
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PVP^{-1} =
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\begin{pmatrix}
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0 & A\\
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B & C
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\end{pmatrix}, \quad A, C, C \in M_{g \times g} (\mathbb{Z})
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\end{align}
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\end{theorem}
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In other words you can find rank $g$ direct summand $\mathcal{Z}$ of $H_1(F)$ \\
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????????????\\
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such that for any
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$\alpha, \beta \in \mathcal{L}$ the linking number $\Lk (\alpha, \beta^+) = 0$.
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\begin{definition}
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An abstract Seifert matrix (i. e.
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\end{definition}
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Choose a basis $(b_1, ..., b_i)$ \\
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???\\
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of $H_2(Y, \mathbb{Z}$, then $A = (b_i, b_y)$ \\??\\ is a matrix of intersection form:
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\begin{align*}
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\quot{\mathbb{Z}^n}{A\mathbb{Z}^n} \cong H_1(Y, \mathbb{Z}).
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\end{align*}
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In particular $\vert \det A\vert = \# H_1(Y, \mathbb{Z})$.\\
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That means - what is happening on boundary is a measure of degeneracy.
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\begin{center}
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\begin{tikzcd}
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[
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column sep=tiny,
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row sep=small,
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ar symbol/.style =%
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{draw=none,"\textstyle#1" description,sloped},
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isomorphic/.style = {ar symbol={\cong}},
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]
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H_1(Y, \mathbb{Z}) &
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\times \quad H_1(Y, \mathbb{Z})&
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\longrightarrow &
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\quot{\mathbb{Q}}{\mathbb{Z}}
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\text{ - a linking form}
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\\
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\quot{\mathbb{Z}^n}{A\mathbb{Z}} \ar[u,isomorphic] &
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\quot{\mathbb{Z}^n}{A\mathbb{Z}} \ar[u,isomorphic] &\\
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\end{tikzcd}
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$(a, b) \mapsto aA^{-1}b^T$
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\end{center}
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?????????????????????????????????\\
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\noindent
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The intersection form on a four-manifold determines the linking on the boundary. \\
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\noindent
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Let $K \in S^1$ be a knot, $\Sigma(K)$ its double branched cover. If $V$ is a Seifert matrix for $K$, then
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$H_1(\Sigma(K), \mathbb{Z}) \cong \quot{\mathbb{Z}^n}{A\mathbb{Z}}$ where
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$A = V \times V^T$, $n = \rank V$.
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%\input{ink_diag}
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\begin{figure}[h]
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\fontsize{20}{10}\selectfont
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\centering{
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\def\svgwidth{\linewidth}
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\resizebox{0.5\textwidth}{!}{\input{images/ball_4.pdf_tex}}
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\caption{Pushing the Seifert surface in 4-ball.}
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\label{fig:pushSeifert}
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}
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\end{figure}
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\noindent
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Let $X$ be the four-manifold obtained via the double branched cover of $B^4$ branched along $\widetilde{\Sigma}$.
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\begin{fact}
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\begin{itemize}
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\item $X$ is a smooth four-manifold,
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\item $H_1(X, \mathbb{Z}) =0$,
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\item $H_2(X, \mathbb{Z}) \cong \mathbb{Z}^n$
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\item The intersection form on $X$ is $V + V^T$.
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\end{itemize}
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\end{fact}
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\begin{figure}[h]
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\fontsize{20}{10}\selectfont
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\centering{
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\def\svgwidth{\linewidth}
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\resizebox{0.5\textwidth}{!}{\input{images/ball_4_pushed_cycle.pdf_tex}}
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\caption{Cycle pushed in 4-ball.}
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\label{fig:pushCycle}
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}
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\end{figure}
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\noindent
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Let $Y = \Sigma(K)$. Then:
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\begin{align*}
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H_1(Y, \mathbb{Z}) \times H_1(Y, \mathbb{Z}) &\longrightarrow \quot{\mathbb{Q}}{\mathbb{Z}}
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\\
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(a,b) &\mapsto a A^{-1} b^{T},\qquad
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A = V + V^T.
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\end{align*}
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????????????????????????????
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\begin{align*}
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H_1(Y, \mathbb{Z}) \cong \quot{\mathbb{Z}^n}{A\mathbb{Z}}\\
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A \longrightarrow BAC^T \quad \text{Smith normal form}
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\end{align*}
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???????????????????????\\
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In general
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%no lecture at 29.04
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\section{\hfill\DTMdate{2019-05-20}}
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Let $M$ be compact, oriented, connected four-dimensional manifold. 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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\begin{center}
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\begin{tikzcd}
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[
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column sep=tiny,
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row sep=small,
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ar symbol/.style = {draw=none,"\textstyle#1" description,sloped},
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isomorphic/.style = {ar symbol={\cong}},
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]
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H_2(M, \mathbb{Z})&
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\times & H_2(M, \mathbb{Z})
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\longrightarrow &
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\mathbb{Z}
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\\
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\ar[u,isomorphic] \mathbb{Z}^n && &\\
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\end{tikzcd}
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\end{center}
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\noindent
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Let us consider a specific case: $M$ has a boundary $Y = \partial M$.
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Betti number $b_1(Y) = 0$, $H_1(Y, \mathbb{Z})$ is finite.
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Then the intersection form can be degenerated in the sense that:
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\begin{align*}
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H_2(M, \mathbb{Z})
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\times H_2(M, \mathbb{Z})
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&\longrightarrow
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\mathbb{Z} \quad&
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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} \quad&
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a &\mapsto (a, \_) H_2(M, \mathbb{Z})
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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, $X = S^3 \setminus K$ a knot complement and
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$\widetilde{X} \xrightarrow{\enspace \rho \enspace} X$ an infinite cyclic cover (universal abelian cover). By Hurewicz theorem we know that:
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\begin{align*}
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\pi_1(X) \longrightarrow \quot{\pi_1(X)}{[\pi_1(X), \pi_1(X)]} = H_1(X, \mathbb{Z} ) \cong \mathbb{Z}
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\end{align*}
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????????????????????????????????????????????????????????????????????????\\
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????????????????????????????????????????????????????????????????????????\\
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????????????????????????????????????????????????????????????????????????\\
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????????????????????????????????????????????????????????????????????????\\
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$C_{*}(\widetilde{X})$ has a structure of a $\mathbb{Z}[t, t^{-1}] \cong \mathbb{Z}[\mathbb{Z}]$ module. \\
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Let $H_1(\widetilde{X}, \mathbb{Z}[t, t^{-1}])$ be the Alexander module of the knot $K$ with an intersection form:
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\begin{align*}
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H_1(\widetilde{X}, \mathbb{Z}[t, t^{-1}]) \times
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H_1(\widetilde{X}, \mathbb{Z}[t, t^{-1}]) \longrightarrow \quot{\mathbb{Q}}{\mathbb{Z}[t, t^{-1}]}
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\end{align*}
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\begin{fact}
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\begin{align*}
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&H_1(\widetilde{X}, \mathbb{Z}[t, t^{-1}]) \cong
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\quot{\mathbb{Z}{[t, t^{-1}]}^n}{(tV - V^T)\mathbb{Z}[t, t^{-1}]^n}\;, \\
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&\text{where $V$ is a Seifert matrix.}
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\end{align*}
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\end{fact}
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\begin{fact}
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\begin{align*}
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H_1(\widetilde{X}, \mathbb{Z}[t, t^{-1}]) \times
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H_1(\widetilde{X}, \mathbb{Z}[t, t^{-1}]) &\longrightarrow \quot{\mathbb{Q}}{\mathbb{Z}[t, t^{-1}]}\\
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(\alpha, \beta) \quad &\mapsto \alpha^{-1}(t -1)(tV - V^T)^{-1}\beta
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\end{align*}
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\end{fact}
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\noindent
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Note that $\mathbb{Z}$ is not PID.
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Therefore we don't have primary decomposition of this module.
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We can simplify this problem by replacing $\mathbb{Z}$ by $\mathbb{R}$. We lose some date by doing this transition, but we can
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\begin{align*}
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\xi \in S^1 \setminus \{ \pm 1\}
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&\quad
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p_{\xi} =
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(t - \xi)(t - \xi^{-1}) t^{-1}
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\\
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\xi \in \mathbb{R} \setminus \{ \pm 1\}
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&\quad
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q_{\xi} = (t - \xi)(t - \xi^{-1}) t^{-1}
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\\
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\xi \notin \mathbb{R} \cup S^1
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&\quad
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q_{\xi} = (t - \xi)(t - \overbar{\xi})(t - \xi^{-1})
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(t - \overbar{\xi}^{-1}) t^{-2}
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\end{align*}
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Let $\Lambda = \mathbb{R}[t, t^{-1}]$. Then:
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\begin{align*}
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H_1(\widetilde{X}, \Lambda) \cong \bigoplus_{\substack{\xi \in S^1 \setminus \{\pm 1 \}\\ k\geq 0}}
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( \quot{\Lambda}{p_{\xi}^k })^{n_k, \xi}
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\oplus
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\bigoplus_{\substack{\xi \notin S^1 \\ l\geq 0}}
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(\quot{\Lambda}{q_{\xi}^l})^{n_l, \xi}&
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\end{align*}
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We can make this composition orthogonal with respect to the Blanchfield paring.
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\vspace{0.5cm}\\
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Historical remark:
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\begin{itemize}
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\item John Milnor, \textit{On isometries of inner product spaces}, 1969,
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\item Walter Neumann, \textit{Invariants of plane curve singularities}
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%in: Knots, braids and singulari- ties (Plans-sur-Bex, 1982), 223–232, Monogr. Enseign. Math., 31, Enseignement Math., Geneva
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, 1983,
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\item András Némethi, \textit{The real Seifert form and the spectral pairs of isolated hypersurfaceenumerate singularities}, 1995,
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%Compositio Mathematica, Volume 98 (1995) no. 1, p. 23-41
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\item Maciej Borodzik, Stefan Friedl
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\textit{The unknotting number and classical invariants II}, 2014.
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\end{itemize}
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\vspace{0.5cm}
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Let $p = p_{\xi}$, $k\geq 0$.
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\begin{align*}
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\quot{\Lambda}{p^k \Lambda} \times
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\quot{\Lambda}{p^k \Lambda} &\longrightarrow \quot{\mathbb{Q}(t)}{\Lambda}\\
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(1, 1) &\mapsto \kappa\\
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\text{Now: } (p^k \cdot 1, 1) &\mapsto 0\\
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p^k \kappa = 0 &\in \quot{\mathbb{Q}(t)}{\Lambda}\\
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\text{therfore } p^k \kappa &\in \Lambda\\
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\text{we have } (1, 1) &\mapsto \frac{h}{p^k}\\
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\end{align*}
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$h$ is not uniquely defined: $h \rightarrow h + g p^k$ doesn't affect paring. \\
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Let $h = p^k \kappa$.
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\begin{example}
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\begin{align*}
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\phi_0 ((1, 1))=\frac{+1}{p}\\
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\phi_1 ((1, 1)) = \frac{-1}{p}
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\end{align*}
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$\phi_0$ and $\phi_1$ are not isomorphic.
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\end{example}
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\begin{proof}
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Let $\Phi:
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\quot{\Lambda}{p^k \Lambda} \longrightarrow
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\quot{\Lambda}{p^k \Lambda}$
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be an isomorphism. \\
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Let: $\Phi(1) = g \in \lambda$
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\begin{align*}
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\quot{\Lambda}{p^k \Lambda}
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\xrightarrow{\enspace \Phi \enspace}&
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\quot{\Lambda}{p^k \Lambda}\\
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\phi_0((1, 1)) = \frac{1}{p^k} \qquad&\qquad
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\phi_1((g, g)) = \frac{1}{p^k} \quad \text{($\Phi$ is an isometry).}
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\end{align*}
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Suppose for the paring $\phi_1((g, g))=\frac{1}{p^k}$ we have $\phi_1((1, 1)) = \frac{-1}{p^k}$. Then:
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\begin{align*}
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\frac{-g\overbar{g}}{p^k} = \frac{1}{p^k} &\in \quot{\mathbb{Q}(t)}{\Lambda}\\
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\frac{-g\overbar{g}}{p^k} - \frac{1}{p^k} &\in \Lambda \\
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-g\overbar{g} &\equiv 1\pmod{p} \text{ in } \Lambda\\
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-g\overbar{g} - 1 &= p^k \omega \text{ for some } \omega \in \Lambda\\
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\text{evalueting at $\xi$: }\\
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\overbrace{-g(\xi)g(\xi^{-1})}^{>0} - 1 = 0 \quad \contradiction
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\end{align*}
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\end{proof}
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????????????????????\\
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\begin{align*}
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g &= \sum{g_i t^i}\\
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\overbar{g} &= \sum{g_i t^{-i}}\\
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\overbar{g}(\xi) &= \sum g_i \xi^i \quad \xi \in S^1\\
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\overbar{g}(\xi) &=\overbar{g(\xi)}
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\end{align*}
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Suppose $g = (t - \xi)^{\alpha} g^{\prime}$. Then $(t - \xi)^{k - \alpha}$ goes to $0$ in $\quot{\Lambda}{p^k \Lambda}$.
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\begin{theorem}
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Every sesquilinear non-degenerate pairing
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\begin{align*}
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\quot{\Lambda}{p^k} \times \quot{\Lambda}{p}
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\longleftrightarrow \frac{h}{p^k}
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\end{align*}
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is isomorphic either to the pairing wit $h=1$ or to the paring with $h=-1$ depending on sign of $h(\xi)$ (which is a real number).
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\end{theorem}
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\begin{proof}
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There are two steps of the proof:
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\begin{enumerate}
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\item
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Reduce to the case when $h$ has a constant sign on $S^1$.
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\item
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Prove in the case, when $h$ has a constant sign on $S^1$.
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\end{enumerate}
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\begin{lemma}
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If $P$ is a symmetric polynomial such that $P(\eta)\geq 0$ for all $\eta \in S^1$, then $P$ can be written as a product $P = g \overbar{g}$ for some polynomial $g$.
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\end{lemma}
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\begin{proof}[Sketch of proof]
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Induction over $\deg P$.\\
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Let $\zeta \notin S^1$ be a root of $P$, $P \in \mathbb{R}[t, t^{-1}]$. Assume $\zeta \notin \mathbb{R}$. We know that polynomial $P$ is divisible by
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$(t - \zeta)$, $(t - \overbar{\zeta})$, $(t^{-1} - \zeta)$ and $(t^{-1} - \overbar{\zeta})$.
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Therefore:
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\begin{align*}
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&P^{\prime} = \frac{P}{(t - \zeta)(t - \overbar{\zeta})(t^{-1} - \zeta)(t^{-1} - \overbar{\zeta})}\\
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&P^{\prime} = g^{\prime}\overbar{g}
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\end{align*}
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We set $g = g^{\prime}(t - \zeta)(t - \overbar{\zeta})$ and
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$P = g \overbar{g}$. Suppose $\zeta \in S^1$. Then $(t - \zeta)^2 \vert P$ (at least - otherwise it would change sign). Therefore:
|
||
\begin{align*}
|
||
&P^{\prime} = \frac{P}{(t - \zeta)^2(t^{-1} - \zeta)^2}\\
|
||
&g = (t - \zeta)(t^{-1} - \zeta) g^{\prime} \quad \text{etc.}
|
||
\end{align*}
|
||
The map $(1, 1) \mapsto \frac{h}{p^k} = \frac{g\overbar{g}h}{p^k}$ is isometric whenever $g$ is coprime with $P$.
|
||
\end{proof}
|
||
\begin{lemma}\label{L:coprime polynomials}
|
||
Suppose $A$ and $B$ are two symmetric polynomials that are coprime and that $\forall z \in S^1$ either $A(z) > 0$ or $B(z) > 0$. Then there exist
|
||
symmetric polynomials $P$, $Q$ such that
|
||
$P(z), Q(z) > 0$ for $z \in S^1$ and $PA + QB \equiv 1$.
|
||
\end{lemma}
|
||
\begin{proof}[Idea of proof]
|
||
For any $z$ find an interval $(a_z, b_z)$ such that if $P(z) \in (a_z, b_z)$ and $P(z)A(z) + Q(z)B(z) = 1$, then $Q(z) > 0$, $x(z) = \frac{az + bz}{i}$ is a continues function on $S^1$ approximating $z$ by a polynomial .
|
||
\\??????????????????????????\\
|
||
\begin{flalign*}
|
||
(1, 1) \mapsto \frac{h}{p^k} \mapsto \frac{g\overbar{g}h}{p^k}&\\
|
||
g\overbar{g} h + p^k\omega = 1&
|
||
\end{flalign*}
|
||
Apply Lemma \ref{L:coprime polynomials} for $A=h$, $B=p^{2k}$. Then, if the assumptions are satisfied,
|
||
\begin{align*}
|
||
Ph + Qp^{2k} = 1\\
|
||
p>0 \Rightarrow p = g \overbar{g}\\
|
||
p = (t - \xi)(t - \overbar{\xi})t^{-1}\\
|
||
\text{so } p \geq 0 \text{ on } S^1\\
|
||
p(t) = 0 \Leftrightarrow
|
||
t = \xi or t = \overbar{\xi}\\
|
||
h(\xi) > 0\\
|
||
h(\overbar{\xi})>0\\
|
||
g\overbar{g}h + Qp^{2k} = 1\\
|
||
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.
|
||
\end{proof}
|
||
?????????????????\\
|
||
\begin{align*}
|
||
(\quot{\Lambda}{p_{\xi}^k} \times
|
||
\quot{\Lambda}{p_{\xi}^k}) &\longrightarrow
|
||
\frac{\epsilon}{p_{\xi}^k}, \quad \xi \in S^1 \setminus\{\pm 1\}\\
|
||
(\quot{\Lambda}{q_{\xi}^k} \times
|
||
\quot{\Lambda}{q_{\xi}^k}) &\longrightarrow
|
||
\frac{1}{q_{\xi}^k}, \quad \xi \notin S^1\\
|
||
\end{align*}
|
||
??????????????????? 1 ?? epsilon?\\
|
||
\begin{theorem}(Matumoto, Borodzik-Conway-Politarczyk)
|
||
Let $K$ be a knot,
|
||
\begin{align*}
|
||
&H_1(\widetilde{X}, \Lambda) \times
|
||
H_1(\widetilde{X}, \Lambda)
|
||
= \bigoplus_{\substack{k, \xi, \epsilon\\ \xi in S^1}}
|
||
(\quot{\Lambda}{p_{\xi}^k}, \epsilon)^{n_k, \xi, \epsilon} \oplus \bigoplus_{k, \eta}
|
||
(\quot{\Lambda}{p_{\xi}^k})^{m_k}
|
||
\end{align*}
|
||
\begin{align*}
|
||
\text{Let } \delta_{\sigma}(\xi) = \lim_{\varepsilon \rightarrow 0^{+}}
|
||
\sigma(e^{2\pi i \varepsilon} \xi)
|
||
- \sigma(e^{-2\pi i \varepsilon} \xi),\\
|
||
\text{then }
|
||
\sigma_j(\xi) = \sigma(\xi) - \frac{1}{2} \lim_{\varepsilon \rightarrow 0}
|
||
\sigma(e^{2\pi i \varepsilon}\xi)
|
||
+ \sigma(e^{-2 \pi i \varepsilon}\xi)
|
||
\end{align*}
|
||
The jump at $\xi$ is equal to
|
||
$2 \sum\limits_{k_i \text{ odd}} \epsilon_i$. The peak of the signature function is equal to $\sum\limits_{k_i \text{even}} \epsilon_i$.
|
||
%$(\eta_{k, \xi_l^{+}} -\eta_{k, \xi_l^{-}}$
|
||
\end{theorem}
|
||
\end{proof}
|
||
\section{\hfill\DTMdate{2019-05-27}}
|
||
|
||
....
|
||
\begin{definition}
|
||
A square hermitian matrix $A$ of size $n$ with coefficients in \\
|
||
the Blanchfield pairing if:
|
||
$H_1(\bar{X}$
|
||
\end{definition}
|
||
|
||
field of fractions
|
||
|
||
\section{\hfill\DTMdate{2019-06-03}}
|
||
\begin{theorem}
|
||
Let $K$ be a knot and $u(K)$ its unknotting number. Let $g_4$ be a minimal four genus of a smooth surface $S$ in $B^4$ such that $\partial S = K$. Then:
|
||
\[
|
||
u(K) \geq g_4(K)
|
||
\]
|
||
\begin{proof}
|
||
Recall that if $u(K)=u$ then $K$ bounds a disk $\Delta$ with $u$ ordinary double points.
|
||
\\
|
||
\noindent
|
||
Remove from $\Delta$ the two self intersecting and glue the Seifert surface for the Hopf link. The reality surface $S$ has Euler characteristic $\chi(S) = 1 - 2u$. Therefore $g_4(S) = u$ .
|
||
\end{proof}
|
||
???????????????????\\
|
||
\begin{example}
|
||
The knot $8_{20}$ is slice: $\sigma \equiv 0$ almost everywhere but $\sigma(e^{\frac{ 2\pi i}{6}}) = + 1$.
|
||
\end{example}
|
||
%ref Structure in the classical knot concordance group
|
||
%Tim D. Cochran, Kent E. Orr, Peter Teichner
|
||
%Journal-ref: Comment. Math. Helv. 79 (2004) 105-123
|
||
\subsection*{Surgery}
|
||
%Rolfsen, geometric group theory, Diffeomorpphism of a torus, Mapping class group
|
||
Recall that $H_1(S^1 \times S^1, \mathbb{Z}) = \mathbb{Z}^3$. As generators for $H_1$ we can set ${\alpha = [S^1 \times \pt]}$ and ${\beta=[\pt \times S^1]}$. Suppose ${\phi: S^1 \times S^1 \longrightarrow S^1 \times S^1}$ is a diffeomorphism.
|
||
Consider an induced map on homology group:
|
||
\begin{align*}
|
||
H_1(S^1 \times S^1, \mathbb{Z}) \ni \phi_* (\alpha) &= p\alpha + q \beta, \quad p, q \in \mathbb{Z},\\
|
||
\phi_*(\beta) &= r \alpha + s \beta, \quad r, s \in \mathbb{Z}, \\
|
||
\phi_* &=
|
||
\begin{pmatrix}
|
||
p & q\\
|
||
r & s
|
||
\end{pmatrix}
|
||
\end{align*}
|
||
As $\phi_*$ is diffeomorphis, it must be invertible over $\mathbb{Z}$. Then for a direction preserving diffeomorphism we have $\det \phi_* = 1$. Therefore $\phi_* \in \Sl(2, \mathbb{Z})$.
|
||
\end{theorem}
|
||
|
||
\vspace{10cm}
|
||
\begin{theorem}
|
||
Every such a matrix can be realized as a torus.
|
||
\end{theorem}
|
||
\begin{proof}
|
||
\begin{enumerate}[label={(\Roman*)}]
|
||
\item
|
||
Geometric reason
|
||
\begin{align*}
|
||
\phi_t:
|
||
S^1 \times S^1 &\longrightarrow S^1 \times S^1 \\
|
||
S^1 \times \pt &\longrightarrow \pt \times S^1 \\
|
||
\pt \times S^1 &\longrightarrow S^1 \times \pt \\
|
||
(x, y) & \mapsto (-y, x)
|
||
\end{align*}
|
||
\item
|
||
\end{enumerate}
|
||
\end{proof}
|
||
|
||
|
||
|
||
|
||
\section{balagan}
|
||
|
||
\noindent
|
||
\noindent
|
||
|
||
\section{\hfill\DTMdate{2019-05-06}}
|
||
|
||
\begin{definition}
|
||
Let $X$ be a knot complement.
|
||
Then $H_1(X, \mathbb{Z}) \cong \mathbb{Z}$ and there exists an epimorphism
|
||
$\pi_1(X) \overset{\phi}\twoheadrightarrow \mathbb{Z}$.\\
|
||
The infinite cyclic cover of a knot complement $X$ is the cover associated with the epimorphism $\phi$.
|
||
\[
|
||
\widetilde{X} \longtwoheadrightarrow X
|
||
\]
|
||
\end{definition}
|
||
%Rolfsen, bachalor thesis of Kamila
|
||
\begin{figure}[h]
|
||
\fontsize{10}{10}\selectfont
|
||
\centering{
|
||
\def\svgwidth{\linewidth}
|
||
\resizebox{1\textwidth}{!}{\input{images/covering.pdf_tex}}
|
||
\caption{Infinite cyclic cover of a knot complement.}
|
||
\label{fig:covering}
|
||
}
|
||
\end{figure}
|
||
\begin{figure}[h]
|
||
\fontsize{10}{10}\selectfont
|
||
\centering{
|
||
\def\svgwidth{\linewidth}
|
||
\resizebox{0.8\textwidth}{!}{\input{images/knot_complement.pdf_tex}}
|
||
\caption{A knot complement.}
|
||
\label{fig:complement}
|
||
}
|
||
\end{figure}
|
||
\noindent
|
||
Formal sums $\sum \phi_i(t) a_i + \sum \phi_j(t)\alpha_j$ \\
|
||
finitely generated as a $\mathbb{Z}[t, t^{-1}]$ module.
|
||
\\
|
||
Let $v_{ij} = \Lk(a_i, a_j^+)$. Then
|
||
$V = \{ v_ij\}_{i, j = 1}^n$ is the Seifert matrix associated to the surface $\Sigma$ and the basis $a_1, \dots, a_n$. Therefore $a_k^+ = \sum_{j} v_{jk} \alpha_j$. Then
|
||
$\Lk(a_i, a_k^+)= \Lk(a_k^+, a_i) = \sum_j v_{jk} \Lk(\alpha_j, a_i) = v_{ik}$.
|
||
We also notice that $\Lk(a_i, a_j^-) = \Lk(a_i^+, a_j)= v_{ij}$ and
|
||
$a_j^- = \sum_k v_{kj} t^{-1} \alpha_j$.
|
||
\\
|
||
\noindent
|
||
The homology of $\widetilde{X}$ is generated by $a_1, \dots, a_n$ and relations.
|
||
\begin{definition}
|
||
The Nakanishi index of a knot is the minimal number of generators of $H_1(\widetilde{X})$.
|
||
\end{definition}
|
||
%see Maciej page
|
||
\noindent
|
||
Remark about notation: sometimes one writes $H_1(X; \mathbb{Z}[t, t^{-1}])$ (what is also notation for twisted homology) instead of $H_1(\widetilde{X})$.
|
||
\\
|
||
?????????????????????
|
||
\\
|
||
\noindent
|
||
$\Sigma_?(K) \rightarrow S^3$ ?????\\
|
||
$H_1(\Sigma_?(K), \mathbb{Z}) = h$\\
|
||
$H \times H \longrightarrow \quot{\mathbb{Q}}{\mathbb{Z}}$\\
|
||
...\\
|
||
|
||
Let now $H = H_1(\widetilde{X})$. Can we define a paring? \\
|
||
Let $c, d \in H(\widetilde{X})$ (see Figure \ref{fig:covering_pairing}), $\Delta$ an Alexander polynomial. We know that $\Delta c = 0 \in H_1(\widetilde{X})$ (Alexander polynomial annihilates all possible elements). Let consider a surface $F$ such that $\partial F = c$. Now consider intersection points $F \cdot d$. This points can exist in any $N_k$ or $S_k$.
|
||
\[
|
||
\frac{1}{\Delta} \sum_{j\in \mathbb{Z} t^{-j}}(F \cdot t^j d) \in \quot{\mathbb{Q}[t, t^{-1}]}{\mathbb{Z}[t, t^{-1}]}
|
||
\]
|
||
\\
|
||
?????????????\\
|
||
\begin{figure}[h]
|
||
\fontsize{10}{10}\selectfont
|
||
\centering{
|
||
\def\svgwidth{\linewidth}
|
||
\resizebox{1\textwidth}{!}{\input{images/covering_pairing.pdf_tex}}
|
||
\caption{$c, d \in H_1(\widetilde{X})$.}
|
||
\label{fig:covering_pairing}
|
||
}
|
||
\end{figure}
|
||
|
||
|
||
\begin{definition}
|
||
The $\mathbb{Z}[t, t^{-1}]$ module $H_1(\widetilde{X})$ is called the Alexander module of knot $K$.
|
||
\end{definition}
|
||
\noindent
|
||
Let $R$ be a PID, $M$ a finitely generated $R$ module. Let us consider
|
||
\[
|
||
R^k \overset{A} \longrightarrow R^n \longtwoheadrightarrow M,
|
||
\]
|
||
where $A$ is a $k \times n$ matrix, assume $k\ge n$. The order of $M$ is the $\gcd$ of all determinants of the $n \times n$ minors of $A$. If $k = n$ then $\ord M = \det A$.
|
||
\begin{theorem}
|
||
Order of $M$ doesn't depend on $A$.
|
||
\end{theorem}
|
||
\noindent
|
||
For knots the order of the Alexander module is the Alexander polynomial.
|
||
\begin{theorem}
|
||
\[
|
||
\forall x \in M: (\ord M) x = 0.
|
||
\]
|
||
\end{theorem}
|
||
\noindent
|
||
$M$ is well defined up to a unit in $R$.
|
||
\subsection*{Blanchfield pairing}
|
||
\section{balagan}
|
||
|
||
\begin{fact}[Milnor Singular Points of Complex Hypersurfaces]
|
||
\end{fact}
|
||
%\end{comment}
|
||
\noindent
|
||
An oriented knot is called negative amphichiral if the mirror image $m(K)$ of $K$ is equivalent the reverse knot of $K$: $K^r$. \\
|
||
\begin{problem}
|
||
Prove that if $K$ is negative amphichiral, then $K \# K = 0$ in
|
||
$\mathscr{C}$.
|
||
%
|
||
%\\
|
||
%Hint: $ -K = m(K)^r = (K^r)^r = K$
|
||
\end{problem}
|
||
\begin{example}
|
||
Figure 8 knot is negative amphichiral.
|
||
\end{example}
|
||
%
|
||
%
|
||
\begin{theorem}
|
||
Let $H_p$ be a $p$ - torsion part of $H$. There exists an orthogonal decomposition of $H_p$:
|
||
\[
|
||
H_p = H_{p, 1} \oplus \dots \oplus H_{p, r_p}.
|
||
\]
|
||
$H_{p, i}$ is a cyclic module:
|
||
\[
|
||
H_{p, i} = \quot{\mathbb{Z}[t, t^{-1}]}{p^{k_i} \mathbb{Z} [t, t^{-1}]}
|
||
\]
|
||
\end{theorem}
|
||
\noindent
|
||
The proof is the same as over $\mathbb{Z}$.
|
||
\noindent
|
||
%Add NotePrintSaveCiteYour opinionEmailShare
|
||
%Saveliev, Nikolai
|
||
|
||
%Lectures on the Topology of 3-Manifolds
|
||
%An Introduction to the Casson Invariant
|
||
|
||
\begin{figure}[h]
|
||
\fontsize{10}{10}\selectfont
|
||
\centering{
|
||
\def\svgwidth{\linewidth}
|
||
\resizebox{0.5\textwidth}{!}{\input{images/ball_4_alpha_beta.pdf_tex}}
|
||
}
|
||
%\caption{Sketch for Fact %%\label{fig:concordance_m}
|
||
\end{figure}
|
||
|
||
\end{document}
|
||
|