marchwicka/lectures_on_knot_theory
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Maria Marchwicka 2019-06-08 13:33:05 +02:00
parent 11a97cecc9
commit d2146a556c
10 changed files with 482 additions and 599 deletions
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images/concordance_sum.pdf
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images/concordance_sum.pdf_tex
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@ -36,7 +36,7 @@
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lectures_on_knot_theory.tex
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@ -48,13 +48,16 @@
{\bfseries}{}% {\bfseries}{}%
{\newline}{}% {\newline}{}%
\theoremstyle{break} \theoremstyle{break}
\newtheorem{lemma}{Lemma}[section] \newtheorem{lemma}{Lemma}[section]
\newtheorem{fact}{Fact}[section] \newtheorem{fact}{Fact}[section]
\newtheorem{corollary}{Corollary}[section] \newtheorem{corollary}{Corollary}[section]
\newtheorem{proposition}{Proposition}[section] \newtheorem{proposition}{Proposition}[section]
\newtheorem{example}{Example}[section] \newtheorem{example}{Example}[section]
\newtheorem{problem}{Problem}[section]
\newtheorem{definition}{Definition}[section] \newtheorem{definition}{Definition}[section]
\newtheorem{theorem}{Theorem}[section] \newtheorem{theorem}{Theorem}[section]
\newcommand{\contradiction}{% \newcommand{\contradiction}{%
\ensuremath{{\Rightarrow\mspace{-2mu}\Leftarrow}}} \ensuremath{{\Rightarrow\mspace{-2mu}\Leftarrow}}}
\newcommand*\quot[2]{{^{\textstyle #1}\big/_{\textstyle #2}}} \newcommand*\quot[2]{{^{\textstyle #1}\big/_{\textstyle #2}}}
@ -66,6 +69,10 @@
\newcommand{\sdots}{\smash{\vdots}} \newcommand{\sdots}{\smash{\vdots}}
\AtBeginDocument{\renewcommand{\setminus}{% \AtBeginDocument{\renewcommand{\setminus}{%
\mathbin{\backslash}}} \mathbin{\backslash}}}
@ -73,6 +80,8 @@
\DeclareMathOperator{\Hom}{Hom} \DeclareMathOperator{\Hom}{Hom}
\DeclareMathOperator{\rank}{rank} \DeclareMathOperator{\rank}{rank}
\DeclareMathOperator{\Gl}{Gl} \DeclareMathOperator{\Gl}{Gl}
\DeclareMathOperator{\Lk}{lk}
\titleformat{\section}{\normalfont \fontsize{12}{15} \bfseries}{% \titleformat{\section}{\normalfont \fontsize{12}{15} \bfseries}{%
Lecture\ \thesection}% Lecture\ \thesection}%
@ -263,12 +272,12 @@ Remark: there are knots that admit non isotopic Seifert surfaces of minimal genu
\begin{definition} \begin{definition}
Suppose $\alpha$ and $\beta$ are two simple closed curves in $\mathbb{R}^3$. Suppose $\alpha$ and $\beta$ are two simple closed curves in $\mathbb{R}^3$.
On a diagram $L$ consider all crossings between $\alpha$ and $\beta$. Let $N_+$ be the number of positive crossings, $N_-$ - negative. Then the linking number: $lk(\alpha, \beta) = \frac{1}{2}(N_+ - N_-)$. On a diagram $L$ consider all crossings between $\alpha$ and $\beta$. Let $N_+$ be the number of positive crossings, $N_-$ - negative. Then the linking number: $\Lk(\alpha, \beta) = \frac{1}{2}(N_+ - N_-)$.
\end{definition} \end{definition}
\hfill \hfill
\\ \\
Let $\alpha$ and $\beta$ be two disjoint simple cross curves in $S^3$. Let $\alpha$ and $\beta$ be two disjoint simple cross curves in $S^3$.
Let $\nu(\beta)$ be a tubular neighbourhood of $\beta$. The linking number can be interpreted via first homology group, where $lk(\alpha, \beta)$ is equal to evaluation of $\alpha$ as element of first homology group of the complement of $\beta$: Let $\nu(\beta)$ be a tubular neighbourhood of $\beta$. The linking number can be interpreted via first homology group, where $\Lk(\alpha, \beta)$ is equal to evaluation of $\alpha$ as element of first homology group of the complement of $\beta$:
\[ \[
\alpha \in H_1(S^3 \setminus \nu(\beta), \mathbb{Z}) \cong \mathbb{Z}.\] \alpha \in H_1(S^3 \setminus \nu(\beta), \mathbb{Z}) \cong \mathbb{Z}.\]
@ -306,7 +315,7 @@ where $b_1$ is first Betti number of $\Sigma$.
\subsection{Seifert matrix} \subsection{Seifert matrix}
Let $L$ be a link and $\Sigma$ be an oriented Seifert surface for $L$. Choose a basis for $H_1(\Sigma, \mathbb{Z})$ consisting of simple closed $\alpha_1, \dots, \alpha_n$. Let $L$ be a link and $\Sigma$ be an oriented Seifert surface for $L$. Choose a basis for $H_1(\Sigma, \mathbb{Z})$ consisting of simple closed $\alpha_1, \dots, \alpha_n$.
Let $\alpha_1^+, \dots \alpha_n^+$ be copies of $\alpha_i$ lifted up off the surface (push up along a vector field normal to $\Sigma$). Note that elements $\alpha_i$ are contained in the Seifert surface while all $\alpha_i^+$ are don't intersect the surface. Let $\alpha_1^+, \dots \alpha_n^+$ be copies of $\alpha_i$ lifted up off the surface (push up along a vector field normal to $\Sigma$). Note that elements $\alpha_i$ are contained in the Seifert surface while all $\alpha_i^+$ are don't intersect the surface.
Let $lk(\alpha_i, \alpha_j^+) = \{a_{ij}\}$. Then the matrix $S = \{a_{ij}\}_{i, j =1}^n$ is called a Seifert matrix for $L$. Note that by choosing a different basis we get a different matrix. Let $\Lk(\alpha_i, \alpha_j^+) = \{a_{ij}\}$. Then the matrix $S = \{a_{ij}\}_{i, j =1}^n$ is called a Seifert matrix for $L$. Note that by choosing a different basis we get a different matrix.
\begin{figure}[h] \begin{figure}[h]
\fontsize{20}{10}\selectfont \fontsize{20}{10}\selectfont
@ -581,19 +590,37 @@ g_{\big| \partial D^2} = f_{\big| \partial D^2.}
&F: \mathbb{C}^2 \rightarrow \mathbb{C} \text{ a polynomial} \\ &F: \mathbb{C}^2 \rightarrow \mathbb{C} \text{ a polynomial} \\
&F(0) = 0 &F(0) = 0
\end{align*} \end{align*}
Fact (Milnor Singular Points of Complex Hypersurfaces):
\end{example}
%\end{comment}
An oriented knot is called negative amphichiral if the mirror image $m(K)$ if $K$ is equivalent the reverse knot of $K$. \\ \end{example}
\begin{example}[Problem] ????????????
Prove that if $K$ is negative amphichiral, then $K \# K$ in \\
$\mathbf{C}$ \noindent
as a corollary we see that $K_T^{n, }$ ???? \\
is not slice unless $m=0$.
\begin{theorem}
The map $j: \mathscr{C} \longrightarrow \mathbb{Z}^{\infty}$ is a surjection that maps ${K_n}$ to a linear independent set. Moreover $\mathscr{C} \cong \mathbb{Z}$
\end{theorem}
\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} \end{example}
% %
% %
% %
\section{\hfill\DTMdate{2019-03-18}} \section{Concordance group \hfill\DTMdate{2019-03-18}}
\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 $B^4$.
@ -628,7 +655,7 @@ $K_1 \# K_2 \sim {K_1}^{\prime} \# {K_2}^{\prime}$.
\fontsize{10}{10}\selectfont \fontsize{10}{10}\selectfont
\centering{ \centering{
\def\svgwidth{\linewidth} \def\svgwidth{\linewidth}
\resizebox{0.8\textwidth}{!}{\input{images/concordance_sum.pdf_tex}} \resizebox{1\textwidth}{!}{\input{images/concordance_sum.pdf_tex}}
} }
\caption{Sketch for Fakt \ref{fakt:concordance_connected}.} \caption{Sketch for Fakt \ref{fakt:concordance_connected}.}
\label{fig:concordance_sum} \label{fig:concordance_sum}
@ -639,23 +666,40 @@ $K_1 \# K_2 \sim {K_1}^{\prime} \# {K_2}^{\prime}$.
$K \# m(K) \sim $ the unknot. $K \# m(K) \sim $ the unknot.
\end{fact} \end{fact}
\noindent \noindent
Let $\mathscr{C}$ denote all equivalent classes for knots. $\mathscr{C}$ is a group under taking connected sums, with neutral element (the class defined by) an unknot and inverse element (a class defined by) a mirror image.\\ \begin{theorem}
The figure eight knot is a torsion element in $\mathscr{C}$ ($2K \sim $ the unknot).\\ Let $\mathscr{C}$ denote a set of all equivalent classes for knots and $\{0\}$ denote class of all knots concordant to a trivial knot.
\begin{example}[Problem] $\mathscr{C}$ is a group under taking connected sums. The neutral element in the group is $\{0\}$ and the inverse element of an element $\{K\} \in \mathscr{C}$ is $-\{K\} = \{mK\}$.
Are there in concordance group torsion elements that are not $2$ torsion elements? (open) \end{theorem}
\end{example} \begin{fact}
The figure eight knot is a torsion element in $\mathscr{C}$ ($2K \sim $ the unknot).
\end{fact}
\begin{problem}[open]
Are there in concordance group torsion elements that are not $2$ torsion elements?
\end{problem}
\noindent \noindent
Remark: $K \sim K^{\prime} \Leftrightarrow K \# -K^{\prime}$ is slice. Remark: $K \sim K^{\prime} \Leftrightarrow K \# -K^{\prime}$ is slice.
\\
\\
\noindent
Let $\Omega$ be an oriented \\
???????\\
Suppose $\Sigma$ is a Seifert matrix with an intersection form ${(\alpha, \beta) \mapsto \Lk(\alpha, \beta^+)}$. Suppose $\alpha, \beta \in H_1(\Sigma, \mathbb{Z}$ (i.e. there are cycles). \\
??????????????\\
$\alpha, \beta \in \ker (H_1(\Sigma, \mathbb{Z}) \longrightarrow H_1(\Omega, \mathbb{Z}))$. Then there are two cycles $A, B \in \Omega$ such that $\partial A = \alpha$ and $\partial B = \beta$.
Let $B^+$ be a push off of $B$ in the positive normal direction such that
$\partial B^+ = \beta^+$.
Then
$\Lk(\alpha, \beta^+) = A \cdot B^+$
% %
% %
\\
\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*}
H_2(X, \mathbb{Z}) \xrightarrow{\text{Poincaré duality}} H^2(X, \mathbb{Z} ) \xrightarrow{\text{evaluation}}\Hom(H_2(X, \mathbb{Z}), \mathbb{Z}) H_2(X, \mathbb{Z}) \xrightarrow{\text{Poincar\'e duality}} H^2(X, \mathbb{Z} ) \xrightarrow{\text{evaluation}}\Hom(H_2(X, \mathbb{Z}), \mathbb{Z})
\end{align*} \end{align*}
Intersection form: Intersection form:
$H_2(X, \mathbb{Z}) \times $H_2(X, \mathbb{Z}) \times
@ -749,7 +793,7 @@ In general
\section{\hfill\DTMdate{2019-05-20}} \section{\hfill\DTMdate{2019-05-20}}
Let $M$ be compact, oriented, connected four-dimensional manifold. If $H_1(M, \mathbb{Z}) = 0$ then there exists a 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 $M$: bilinear form - the intersection form on $M$:
\begin{center} \begin{center}
@ -1006,22 +1050,95 @@ field of fractions
\section{balagan} \section{balagan}
\begin{comment} \noindent
\setlength{\arraycolsep}{2em} \begin{proof}
\newcommand{\lbrce}{\smash{\left.\rule{0pt}{25pt}\right\}}} By Poincar\'e duality we know that:
\newcommand{\rbrce}{\smash{\left\{\rule{0pt}{25pt}\right.}} \begin{align*}
H_3(\Omega, Y) &\cong H^0(\Omega),\\
H_2(Y) &\cong H^0(Y),\\
H_2(\Omega) &\cong H^1(\Omega, Y),\\
H_2(\Omega, Y) &\cong H^1(\Omega).
\end{align*}
Therefore $\dim_{\mathbb{Q}} \quot{H_1(Y)}{V}
= \dim_{\mathbb{Q}} V
$.
\end{proof}
\noindent
Suppose $g(K) = 0$ ($K$ is slice). Then $H_1(\Sigma, \mathbb{Z}) \cong H_1(Y, \mathbb{Z})$. Let $g_{\Sigma}$ be the genus of $\Sigma$, $\dim H_1(Y, \mathbb{Z}) = 2g_{\Sigma}$. Then the Seifert form $V$ on 4\\
?????\\
has a subspace of dimension $g_{\Sigma}$ on which it is zero:
\[ \[
\begin{pmatrix} V =
0 & 0 & 0 \\ \begin{pmatrix}
\sdots & \sdots\makebox[0pt][l]{$\lbrce\left\lceil\frac i2\right\rceil$} & \sdots \\ \begin{array}{c|c}
0 & 0 & \\ 0 & * \\
\hline \hline
* & *
& & 0 \\ \end{array}
& & \makebox[0pt][r]{$\left\lfloor\frac i2\right\rfloor\rbrce$}\sdots \\ \end{pmatrix}
0 & & 0
\end{pmatrix}
\] \]
\begin{align*}
\newcommand*{\AddLeft}[1]{%
\vadjust{%
\vbox to 0pt{%
\vss
\llap{$%
{#1}\left\{
\vphantom{
\begin{matrix}1\\\vdots\\0\end{matrix}
}
\right.\kern-\nulldelimiterspace
\kern0.5em
$}%
\kern0pt
}%
}%
}
V = \qquad
\begin{pmatrix}
0 & \cdots & 0 & * & \cdots & * \\
\vdots & & \vdots & \vdots & &\vdots \\
0 & \cdots & 0 & * & \cdots & *
\AddLeft{g_{\Sigma}}\\
* & \cdots & * & * & \cdots & * \\
\vdots & & \vdots & \vdots & &\vdots \\
* & \cdots & * & * & \cdots & *
\end{pmatrix}_{2g_{\Sigma} \times 2g_{\Sigma}}
\end{align*}
\begin{align*}
\newcommand\coolover[2]{\mathrlap{\smash{\overbrace{\phantom{%
\begin{matrix} #2 \end{matrix}}}^{\mbox{$#1$}}}}#2}
\newcommand\coolunder[2]{\mathrlap{\smash{\underbrace{\phantom{%
\begin{matrix} #2 \end{matrix}}}_{\mbox{$#1$}}}}#2}
\newcommand\coolleftbrace[2]{%
#1\left\{\vphantom{\begin{matrix} #2 \end{matrix}}\right.}
\newcommand\coolrightbrace[2]{%
\left.\vphantom{\begin{matrix} #1 \end{matrix}}\right\}#2}
\vphantom{% phantom stuff for correct box dimensions
\begin{matrix}
\overbrace{XYZ}^{\mbox{$R$}}\\ \\ \\ \\ \\ \\
\underbrace{pqr}_{\mbox{$S$}}
\end{matrix}}%
\begin{matrix}% matrix for left braces
\vphantom{a}\\
\coolleftbrace{A}{e \\ y\\ y}\\
\coolleftbrace{B}{y \\i \\ m}
\end{matrix}%
\begin{bmatrix}
a & \coolover{R}{b & c & d} & x & \coolover{Z}{x & x}\\
e & f & g & h & x & x & x \\
y & y & y & y & y & y & y \\
y & y & y & y & y & y & y \\
y & y & y & y & y & y & y \\
i & j & k & l & x & x & x \\
m & \coolunder{S}{n & o} & \coolunder{W}{p & x & x} & x
\end{bmatrix}%
\begin{matrix}% matrix for right braces
\coolrightbrace{x \\ x \\ y\\ y}{T}\\
\coolrightbrace{y \\ y \\ x }{U}
\end{matrix}
\end{align*}
\end{comment}
\end{document} \end{document}