some inkspace pictures
This commit is contained in:
Maria Marchwicka
parent
16f96e3333
commit
b9d956bfef
@ -1,824 +0,0 @@
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{\bfseries}{}%
|
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|
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{\newline}{}%
|
{\newline}{}%
|
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\theoremstyle{break}
|
\theoremstyle{break}
|
||||||
\newtheorem{lemma}{Lemma}
|
\newtheorem{lemma}{Lemma}[section]
|
||||||
\newtheorem{fact}{Fact}
|
\newtheorem{fact}{Fact}[section]
|
||||||
\newtheorem{corollary}{Corollary}
|
\newtheorem{corollary}{Corollary}[section]
|
||||||
\newtheorem{example}{Example}
|
\newtheorem{proposition}{Proposition}[section]
|
||||||
\newtheorem{definition}{Definition}
|
\newtheorem{example}{Example}[section]
|
||||||
\newtheorem{theorem}{Theorem}
|
\newtheorem{definition}{Definition}[section]
|
||||||
\newtheorem{proposition}{Proposition}
|
\newtheorem{theorem}{Theorem}[section]
|
||||||
|
|
||||||
\newcommand{\contradiction}{%
|
\newcommand{\contradiction}{%
|
||||||
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|
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|
||||||
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|
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|
||||||
|
|
||||||
\newcommand{\overbar}[1]{%
|
\newcommand{\overbar}[1]{%
|
||||||
\mkern 1.5mu=\overline{%
|
\mkern 1.5mu=\overline{%
|
||||||
\mkern-1.5mu#1\mkern-1.5mu}%
|
\mkern-1.5mu#1\mkern-1.5mu}%
|
||||||
\mkern 1.5mu}
|
\mkern 1.5mu}
|
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|
|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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\titleformat{\section}{\normalfont \fontsize{12}{15} \bfseries}{%
|
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|
Lecture\ \thesection}%
|
||||||
|
{2.3ex plus .2ex}{}
|
||||||
|
\titlespacing*{\section}
|
||||||
|
{0pt}{16.5ex plus 1ex minus .2ex}{4.3ex plus .2ex}
|
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|
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|
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\titleformat{\section}{\normalfont \large \bfseries}{%
|
\setlist[itemize]{topsep=0pt,before=%
|
||||||
Lecture\ \thesection}{2.3ex plus .2ex}{}
|
\leavevmode\vspace{0.5em}}
|
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|
|
||||||
%\setlist[itemize]{topsep=0pt,before=%\leavevmode\vspace{0.5em}}
|
|
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|
|
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\input{knots_macros}
|
\input{knots_macros}
|
||||||
@ -87,7 +94,7 @@
|
|||||||
%\newpage
|
%\newpage
|
||||||
%\input{myNotes}
|
%\input{myNotes}
|
||||||
|
|
||||||
\section{\hfill\DTMdate{2019-02-25}}
|
\section{Basic definitions \hfill\DTMdate{2019-02-25}}
|
||||||
\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*}
|
||||||
@ -96,12 +103,13 @@ A knot $K$ in $S^3$ is a smooth (PL - smooth) embedding of a circle $S^1$ in $S^
|
|||||||
\end{definition}
|
\end{definition}
|
||||||
\noindent
|
\noindent
|
||||||
Usually we think about a knot as an image of an embedding: $K = \varphi(S^1)$.
|
Usually we think about a knot as an image of an embedding: $K = \varphi(S^1)$.
|
||||||
|
|
||||||
\begin{example}
|
\begin{example}
|
||||||
\begin{itemize}
|
\begin{itemize}
|
||||||
\item
|
\item
|
||||||
Knots:
|
Knots:
|
||||||
\includegraphics[width=0.08\textwidth]{unknot.png},
|
\includegraphics[width=0.08\textwidth]{unknot.png} (unknot),
|
||||||
\includegraphics[width=0.08\textwidth]{trefoil.png}.
|
\includegraphics[width=0.08\textwidth]{trefoil.png} (trefoil).
|
||||||
\item
|
\item
|
||||||
Not knots:
|
Not knots:
|
||||||
\includegraphics[width=0.12\textwidth]{not_injective_knot.png}
|
\includegraphics[width=0.12\textwidth]{not_injective_knot.png}
|
||||||
@ -172,7 +180,7 @@ Let $D$ be a diagram of an oriented link (to each component of a link we add an
|
|||||||
We can distinguish two types of crossings: right-handed
|
We can distinguish two types of crossings: right-handed
|
||||||
$\left(\PICorientpluscross\right)$, called a positive crossing, and left-handed $\left(\PICorientminuscross\right)$, called a negative crossing.
|
$\left(\PICorientpluscross\right)$, called a positive crossing, and left-handed $\left(\PICorientminuscross\right)$, called a negative crossing.
|
||||||
|
|
||||||
\section*{Reidemeister moves}
|
\subsection{Reidemeister moves}
|
||||||
A Reidemeister move is one of the three types of operation on a link diagram as shown below:
|
A Reidemeister move is one of the three types of operation on a link diagram as shown below:
|
||||||
\begin{enumerate}[label=\Roman*]
|
\begin{enumerate}[label=\Roman*]
|
||||||
\item\hfill\\
|
\item\hfill\\
|
||||||
@ -197,7 +205,7 @@ deformed into each other by a finite sequence of Reidemeister moves (and isotopy
|
|||||||
%Singularities of Differentiable Maps
|
%Singularities of Differentiable Maps
|
||||||
%Authors: Arnold, V.I., Varchenko, Alexander, Gusein-Zade, S.M.
|
%Authors: Arnold, V.I., Varchenko, Alexander, Gusein-Zade, S.M.
|
||||||
|
|
||||||
\subsection*{Seifert surface}
|
\subsection{Seifert surface}
|
||||||
\noindent
|
\noindent
|
||||||
Let $D$ be an oriented diagram of a link $L$. We change the diagram by smoothing each crossing:
|
Let $D$ be an oriented diagram of a link $L$. We change the diagram by smoothing each crossing:
|
||||||
\begin{align*}
|
\begin{align*}
|
||||||
@ -221,7 +229,7 @@ Note: in general the obtained surface doesn't need to be connected, but by takin
|
|||||||
|
|
||||||
\begin{figure}[H]
|
\begin{figure}[H]
|
||||||
\begin{center}
|
\begin{center}
|
||||||
\includegraphics[width=0.6\textwidth]{seifert_connect.png}
|
\includegraphics[width=0.4\textwidth]{seifert_connect.png}
|
||||||
\end{center}
|
\end{center}
|
||||||
\caption{Connecting two surfaces.}
|
\caption{Connecting two surfaces.}
|
||||||
\label{fig:SeifertConnect}
|
\label{fig:SeifertConnect}
|
||||||
@ -259,13 +267,16 @@ On a diagram $L$ consider all crossings between $\alpha$ and $\beta$. Let $N_+$
|
|||||||
\end{definition}
|
\end{definition}
|
||||||
\hfill
|
\hfill
|
||||||
\\
|
\\
|
||||||
Let $\nu(\beta)$ be a tubular neighbourhood of a closed simple curve $\beta$. 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 in complement of $\beta$ 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 $\beta$:
|
||||||
\[
|
\[
|
||||||
\alpha \in H_1(S^3 \setminus \nu(\beta), \mathbb{Z}) \cong \mathbb{Z}.\]
|
\alpha \in H_1(S^3 \setminus \nu(\beta), \mathbb{Z}) \cong \mathbb{Z}.\]
|
||||||
|
|
||||||
|
|
||||||
\begin{example}
|
\begin{example}
|
||||||
\begin{itemize}\hfill
|
\begin{itemize}
|
||||||
\item
|
\item
|
||||||
Hopf link\hfill
|
Hopf link
|
||||||
\begin{figure}[H]
|
\begin{figure}[H]
|
||||||
\fontsize{20}{10}\selectfont
|
\fontsize{20}{10}\selectfont
|
||||||
\centering{
|
\centering{
|
||||||
@ -274,7 +285,7 @@ Hopf link\hfill
|
|||||||
}
|
}
|
||||||
\end{figure}
|
\end{figure}
|
||||||
\item
|
\item
|
||||||
$T(6, 2)$ link\hfill
|
$T(6, 2)$ link
|
||||||
\begin{figure}[H]
|
\begin{figure}[H]
|
||||||
\fontsize{20}{10}\selectfont
|
\fontsize{20}{10}\selectfont
|
||||||
\centering{
|
\centering{
|
||||||
@ -285,94 +296,263 @@ $T(6, 2)$ link\hfill
|
|||||||
\end{itemize}
|
\end{itemize}
|
||||||
\end{example}
|
\end{example}
|
||||||
|
|
||||||
Let $L$ be a link and $\Sigma$ be a Seifert surface for $L$. Choose a basis for $H_1(\Sigma, \mathbb{Z})$ consisting of simple closed $\alpha_1, \dots, \alpha_n$.
|
\subsection{Seifert matrix}
|
||||||
Let $\alpha_1^+, \dots \alpha_n^+$ be copies of $\alpha_i$ lifted up off the surface. Let $lk(\alpha_i, \alpha_i^+) = \{a_{ij}\}$. Then the matrix $S = \{a_{ij}\}_{i, j =1}^n$ is called a Seifert matrix for $L$.
|
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 $lk(\alpha_i, \alpha_j^+) = \{a_{ij}\}$. 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]
|
||||||
|
\fontsize{20}{10}\selectfont
|
||||||
|
\centering{
|
||||||
|
\def\svgwidth{\linewidth}
|
||||||
|
\resizebox{0.8\textwidth}{!}{\input{images/seifert_matrix.pdf_tex}}
|
||||||
|
}
|
||||||
|
\end{figure}
|
||||||
|
|
||||||
\begin{theorem}
|
\begin{theorem}
|
||||||
The Seifert matrices $S_1$ and $S_2$ for the same link $L$ are S-equivalent, that is, $S_2$ can be obtained from $S_1$ by a sequence of following moves:
|
The Seifert matrices $S_1$ and $S_2$ for the same link $L$ are S-equivalent, that is, $S_2$ can be obtained from $S_1$ by a sequence of following moves:
|
||||||
\begin{enumerate}[label={(\arabic*)}]
|
\begin{enumerate}[label={(\arabic*)}]
|
||||||
|
|
||||||
\item
|
\item
|
||||||
$V \rightarrow AVA^T$ for $A \in $
|
$V \rightarrow AVA^T$, where $A$ is a matrix with integer coefficients,
|
||||||
|
|
||||||
\item
|
\item
|
||||||
|
|
||||||
$V \rightarrow
|
$V \rightarrow
|
||||||
\begin{pmatrix}
|
\begin{pmatrix}
|
||||||
\alpha & * \\
|
|
||||||
\gamma^{*} & \delta
|
|
||||||
\end{pmatrix}
|
|
||||||
$\\
|
|
||||||
\[
|
|
||||||
\begin{pmatrix}
|
|
||||||
\begin{array}{c|c}
|
\begin{array}{c|c}
|
||||||
\epsilon' [T|_A]\epsilon & \ast \\
|
V &
|
||||||
\hline
|
\begin{matrix}
|
||||||
0 & _{\overline{B}'} [\overline{T}]
|
\ast & 0 \\
|
||||||
_{\overline{B}\vphantom{\overline{B}'}}
|
\sdots & \sdots\\
|
||||||
|
\ast & 0
|
||||||
|
\end{matrix} \\
|
||||||
|
\hline
|
||||||
|
\begin{matrix}
|
||||||
|
\ast & \dots & \ast\\
|
||||||
|
0 & \dots & 0
|
||||||
|
\end{matrix}
|
||||||
|
&
|
||||||
|
\begin{matrix}
|
||||||
|
0 & 0\\
|
||||||
|
1 & 0
|
||||||
|
\end{matrix}
|
||||||
\end{array}
|
\end{array}
|
||||||
\end{pmatrix}
|
\end{pmatrix} \quad$
|
||||||
\]\\
|
or
|
||||||
\[\left|
|
$\quad
|
||||||
\begin{array}{cr}
|
V \rightarrow
|
||||||
Q & \begin{matrix} 0 \\ 0 \end{matrix} \\
|
\begin{pmatrix}
|
||||||
\begin{matrix} 2 & 3 \end{matrix} & -1
|
\begin{array}{c|c}
|
||||||
\end{array}
|
V &
|
||||||
\right|\]
|
\begin{matrix}
|
||||||
\\
|
\ast & 0 \\
|
||||||
\[
|
\sdots & \sdots\\
|
||||||
\left[
|
\ast & 0
|
||||||
\begin{array}{c@{}c@{}c}
|
\end{matrix} \\
|
||||||
\left[\begin{array}{cc}
|
\hline
|
||||||
a_{11} & a_{12} \\
|
\begin{matrix}
|
||||||
a_{21} & a_{22} \\
|
\ast & \dots & \ast\\
|
||||||
\end{array}\right] & \mathbf{0} & \mathbf{0} \\
|
0 & \dots & 0
|
||||||
\mathbf{0} & \left[\begin{array}{ccc}
|
\end{matrix}
|
||||||
b_{11} & b_{12} & b_{13}\\
|
&
|
||||||
b_{21} & b_{22} & b_{23}\\
|
\begin{matrix}
|
||||||
b_{31} & b_{32} & b_{33}\\
|
0 & 1\\
|
||||||
\end{array}\right] & \mathbf{0}\\
|
0 & 0
|
||||||
\mathbf{0} & \mathbf{0} & \left[ \begin{array}{cc}
|
\end{matrix}
|
||||||
c_{11} & c_{12} \\
|
\end{array}
|
||||||
c_{21} & c_{22} \\
|
\end{pmatrix}$
|
||||||
\end{array}\right] \\
|
|
||||||
\end{array}\right]
|
|
||||||
\] \\
|
|
||||||
\[
|
|
||||||
\begin{bmatrix}
|
|
||||||
\begin{bmatrix}
|
|
||||||
a_{11} & a_{12}\\
|
|
||||||
a_{21} & a_{22}\\
|
|
||||||
\end{bmatrix} & \mathbf{0} & \mathbf{0} \\
|
|
||||||
\mathbf{0} & \begin{bmatrix}
|
|
||||||
b_{11} & b_{12} & b_{13}\\
|
|
||||||
b_{21} & b_{22} & b_{23}\\
|
|
||||||
b_{31} & b_{32} & b_{33}\\
|
|
||||||
\end{bmatrix} & \mathbf{0} \\
|
|
||||||
\mathbf{0} & \mathbf{0} & \begin{bmatrix}
|
|
||||||
c_{11} & c_{12}\\
|
|
||||||
c_{21} & c_{22}\\
|
|
||||||
\end{bmatrix} \\
|
|
||||||
\end{bmatrix}
|
|
||||||
\]\\
|
|
||||||
\setlength{\arraycolsep}{2em}
|
|
||||||
\newcommand{\lbrce}{\smash{\left.\rule{0pt}{25pt}\right\}}}
|
|
||||||
\newcommand{\rbrce}{\smash{\left\{\rule{0pt}{25pt}\right.}}
|
|
||||||
\newcommand{\sdots}{\smash{\vdots}}
|
|
||||||
\[
|
|
||||||
\begin{pmatrix}
|
|
||||||
0 & 0 & 0 \\
|
|
||||||
\sdots & \sdots\makebox[0pt][l]{$\lbrce\left\lceil\frac i2\right\rceil$} & \sdots \\
|
|
||||||
0 & 0 & \\
|
|
||||||
& & 0 \\
|
|
||||||
& & \makebox[0pt][r]{$\left\lfloor\frac i2\right\rfloor\rbrce$}\sdots \\
|
|
||||||
0 & & 0
|
|
||||||
\end{pmatrix}
|
|
||||||
\]
|
|
||||||
|
|
||||||
\item
|
\item
|
||||||
inverse of (2)
|
inverse of (2)
|
||||||
|
|
||||||
\end{enumerate}
|
\end{enumerate}
|
||||||
\end{theorem}
|
\end{theorem}
|
||||||
|
|
||||||
|
\section{\hfill\DTMdate{2019-03-04}}
|
||||||
|
\begin{theorem}
|
||||||
|
For any knot $K \subset S^3$ there exists a connected, compact and orientable surface $\Sigma(K)$ such that $\partial \Sigma(K) = K$
|
||||||
|
\end{theorem}
|
||||||
|
\begin{proof}("joke")\\
|
||||||
|
Let $K \in S^3$ be a knot and $N = \nu(K)$ be its tubular neighbourhood. Because $K$ and $N$ are homotopy equivalent, we get:
|
||||||
|
\begin{align*}
|
||||||
|
H^1(S^3 \setminus N ) \cong H^1(S^3 \setminus K).
|
||||||
|
\end{align*}
|
||||||
|
Let us consider a long exact sequence of cohomology of a pair $(S^3, S^3 \setminus N)$ with integer coefficients:
|
||||||
|
|
||||||
|
\begin{center}
|
||||||
|
\begin{tikzcd}
|
||||||
|
[
|
||||||
|
column sep=0cm, fill=none,
|
||||||
|
row sep=small,
|
||||||
|
ar symbol/.style =%
|
||||||
|
{draw=none,"\textstyle#1" description,sloped},
|
||||||
|
isomorphic/.style = {ar symbol={\cong}},
|
||||||
|
]
|
||||||
|
&\mathbb{Z}
|
||||||
|
\\
|
||||||
|
|
||||||
|
& H^0(S^3) \ar[u,isomorphic] \to
|
||||||
|
&H^0(S^3 \setminus N) \to
|
||||||
|
\\
|
||||||
|
\to H^1(S^3, S^3 \setminus N) \to
|
||||||
|
& H^1(S^3) \to
|
||||||
|
& H^1(S^3\setminus N) \to
|
||||||
|
\\
|
||||||
|
& 0 \ar[u,isomorphic]&
|
||||||
|
\\
|
||||||
|
\to H^2(S^3, S^3 \setminus N) \to
|
||||||
|
& H^2(S^3) \ar[u,isomorphic] \to
|
||||||
|
& H^2(S^3\setminus N) \to
|
||||||
|
\\
|
||||||
|
\to H^3(S^3, S^3\setminus N)\to
|
||||||
|
& H^3(S) \to
|
||||||
|
& 0
|
||||||
|
\\
|
||||||
|
& \mathbb{Z} \ar[u,isomorphic] &\\
|
||||||
|
\end{tikzcd}
|
||||||
|
\end{center}
|
||||||
|
\[
|
||||||
|
H^* (S^3, S^3 \setminus N) \cong H^* (N, \partial N)
|
||||||
|
\]
|
||||||
|
\\
|
||||||
|
??????????????
|
||||||
|
\\
|
||||||
|
|
||||||
|
\end{proof}
|
||||||
|
|
||||||
|
\begin{definition}
|
||||||
|
Let $S$ be a Seifert matrix for a knot $K$. The Alexander polynomial $\Delta_K(t)$ is a Laurent polynomial:
|
||||||
|
\[
|
||||||
|
\Delta_K(t) := \det (tS - S^T) \in
|
||||||
|
\mathbb{Z}[t, t^{-1}] \cong \mathbb{Z}[\mathbb{Z}]
|
||||||
|
\]
|
||||||
|
\end{definition}
|
||||||
|
|
||||||
|
\begin{theorem}
|
||||||
|
$\Delta_K(t)$ is well defined up to multiplication by $\pm t^k$, for $k \in \mathbb{Z}$.
|
||||||
|
\end{theorem}
|
||||||
|
\begin{proof}
|
||||||
|
We need to show that $\Delta_K(t)$ doesn't depend on $S$-equivalence relation.
|
||||||
|
\begin{enumerate}[label={(\arabic*)}]
|
||||||
|
\item Suppose $S\prime = CSC^T$, $C \in \Gl(n, \mathbb{Z})$ (matrices invertible over $\mathbb{Z}$). Then $\det C = 1$ and:
|
||||||
|
\begin{align*}
|
||||||
|
&\det(tS\prime - S\prime^T) =
|
||||||
|
\det(tCSC^T - (CSC^T)^T) =\\
|
||||||
|
&\det(tCSC^T - CS^TC^T) =
|
||||||
|
\det C(tS - S^T)C^T =
|
||||||
|
\det(tS - S^T)
|
||||||
|
\end{align*}
|
||||||
|
\item
|
||||||
|
Let \\
|
||||||
|
$ A := t
|
||||||
|
\begin{pmatrix}
|
||||||
|
\begin{array}{c|c}
|
||||||
|
S &
|
||||||
|
\begin{matrix}
|
||||||
|
\ast & 0 \\
|
||||||
|
\sdots & \sdots\\
|
||||||
|
\ast & 0
|
||||||
|
\end{matrix} \\
|
||||||
|
\hline
|
||||||
|
\begin{matrix}
|
||||||
|
\ast & \dots & \ast\\
|
||||||
|
0 & \dots & 0
|
||||||
|
\end{matrix}
|
||||||
|
&
|
||||||
|
\begin{matrix}
|
||||||
|
0 & 0\\
|
||||||
|
1 & 0
|
||||||
|
\end{matrix}
|
||||||
|
\end{array}
|
||||||
|
\end{pmatrix}
|
||||||
|
-
|
||||||
|
\begin{pmatrix}
|
||||||
|
\begin{array}{c|c}
|
||||||
|
S^T &
|
||||||
|
\begin{matrix}
|
||||||
|
\ast & 0 \\
|
||||||
|
\sdots & \sdots\\
|
||||||
|
\ast & 0
|
||||||
|
\end{matrix} \\
|
||||||
|
\hline
|
||||||
|
\begin{matrix}
|
||||||
|
\ast & \dots & \ast\\
|
||||||
|
0 & \dots & 0
|
||||||
|
\end{matrix}
|
||||||
|
&
|
||||||
|
\begin{matrix}
|
||||||
|
0 & 1\\
|
||||||
|
0 & 0
|
||||||
|
\end{matrix}
|
||||||
|
\end{array}
|
||||||
|
\end{pmatrix}
|
||||||
|
=
|
||||||
|
\begin{pmatrix}
|
||||||
|
\begin{array}{c|c}
|
||||||
|
tS - S^T &
|
||||||
|
\begin{matrix}
|
||||||
|
\ast & 0 \\
|
||||||
|
\sdots & \sdots\\
|
||||||
|
\ast & 0
|
||||||
|
\end{matrix} \\
|
||||||
|
\hline
|
||||||
|
\begin{matrix}
|
||||||
|
\ast & \dots & \ast\\
|
||||||
|
0 & \dots & 0
|
||||||
|
\end{matrix}
|
||||||
|
&
|
||||||
|
\begin{matrix}
|
||||||
|
0 & -1\\
|
||||||
|
t & 0
|
||||||
|
\end{matrix}
|
||||||
|
\end{array}
|
||||||
|
\end{pmatrix}
|
||||||
|
$
|
||||||
|
\\
|
||||||
|
\\
|
||||||
|
Using the Laplace expansion we get $\det A = \pm t \det(tS - S^T)$.
|
||||||
|
\end{enumerate}
|
||||||
|
\end{proof}
|
||||||
|
%
|
||||||
|
%
|
||||||
|
%
|
||||||
|
\begin{example}
|
||||||
|
If $K$ is a trefoil then we can take
|
||||||
|
$S = \begin{pmatrix}
|
||||||
|
-1 & -1 \\
|
||||||
|
0 & -1
|
||||||
|
\end{pmatrix}$.
|
||||||
|
\[
|
||||||
|
\Delta_K(t) = \det
|
||||||
|
\begin{pmatrix}
|
||||||
|
-t + 1 & -t\\
|
||||||
|
1 & -t +1
|
||||||
|
\end{pmatrix}
|
||||||
|
= (t -1)^2 + t = t^2 - t +1 \ne 1
|
||||||
|
\Rightarrow \text{trefoil is not trivial}
|
||||||
|
\]
|
||||||
|
\end{example}
|
||||||
|
\begin{fact}
|
||||||
|
$\Delta_K(t)$ is symmetric.
|
||||||
|
\end{fact}
|
||||||
|
\begin{proof}
|
||||||
|
Let $S$ be an $n \times n$ matrix.
|
||||||
|
\begin{align*}
|
||||||
|
&\Delta_K(t^{-1}) = \det (t^{-1}S - S^T) = (-t)^{-n} \det(tS^T - S) = \\
|
||||||
|
&(-t)^{-n} \det (tS - S^T) = (-t)^{-n} \Delta_K(t)
|
||||||
|
\end{align*}
|
||||||
|
If $K$ is a knot, then $n$ is necessarily even, and so $\Delta_K(t^{-1}) = t^{-n} \Delta_K(t)$.
|
||||||
|
\end{proof}
|
||||||
|
\begin{lemma}
|
||||||
|
\begin{align*}
|
||||||
|
\frac{1}{2} \deg \Delta_K(t) \leq g_3(K),
|
||||||
|
\text{ where } deg (a_n t^n + \cdots + a_1 t^l )= k - l.
|
||||||
|
\end{align*}
|
||||||
|
\end{lemma}
|
||||||
|
\begin{proof}
|
||||||
|
|
||||||
|
\end{proof}
|
||||||
|
%removing one disk from surface doesn't change $H_1$ (only $H_2$)
|
||||||
\section{}
|
\section{}
|
||||||
\begin{example}
|
\begin{example}
|
||||||
\begin{align*}
|
\begin{align*}
|
||||||
@ -389,17 +569,6 @@ Prove that if $K$ is negative amphichiral, then $K \# K$ in
|
|||||||
$\mathbf{C}$
|
$\mathbf{C}$
|
||||||
\end{example}
|
\end{example}
|
||||||
|
|
||||||
\section{\hfill\DTMdate{2019-03-04}}
|
|
||||||
\begin{proof}("joke")\\
|
|
||||||
Let $K \in S^3$ be a knot and $N$ be its tubular neighbourhood.
|
|
||||||
\begin{align*}
|
|
||||||
H^1(S^3 \setminus N ) \cong H^1(S^3 \setminus K)
|
|
||||||
\end{align*}
|
|
||||||
For a pair $(S^3, S^3 \setminus N)$ we have:
|
|
||||||
\begin{align*}
|
|
||||||
H^0(S^3)
|
|
||||||
\end{align*}
|
|
||||||
\end{proof}
|
|
||||||
\section{\hfill\DTMdate{2019-03-18}}
|
\section{\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. \\
|
||||||
@ -447,7 +616,11 @@ $H_2(X, \mathbb{Z}) \times
|
|||||||
H_2(X, \mathbb{Z}) \longrightarrow \mathbb{Z}$ - 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 $X$.
|
||||||
|
\begin{proposition}
|
||||||
|
$A \cdot B$ doesn't depend of choice of $A$ and $B$ in their homology classes.
|
||||||
|
%$A \cdot B$ gives the pairing as ??
|
||||||
|
|
||||||
|
\end{proposition}
|
||||||
|
|
||||||
\section{\hfill\DTMdate{2019-04-15}}
|
\section{\hfill\DTMdate{2019-04-15}}
|
||||||
In other words:\\
|
In other words:\\
|
||||||
@ -459,27 +632,28 @@ of $H_2(Y, \mathbb{Z}$, then $A = (b_i, b_y)$ \\??\\ is a matrix of intersection
|
|||||||
\end{align*}
|
\end{align*}
|
||||||
In particular $\mid \det A\mid = \# H_1(Y, \mathbb{Z}$.\\
|
In particular $\mid \det A\mid = \# H_1(Y, \mathbb{Z}$.\\
|
||||||
That means - what is happening on boundary is a measure of degeneracy.
|
That means - what is happening on boundary is a measure of degeneracy.
|
||||||
\\
|
|
||||||
\vspace{1cm}
|
|
||||||
\begin{center}
|
\begin{center}
|
||||||
\begin{tikzcd}
|
\begin{tikzcd}
|
||||||
[
|
[
|
||||||
column sep=tiny,
|
column sep=tiny,
|
||||||
row sep=small,
|
row sep=small,
|
||||||
ar symbol/.style = {draw=none,"\textstyle#1" description,sloped},
|
ar symbol/.style =%
|
||||||
isomorphic/.style = {ar symbol={\cong}},
|
{draw=none,"\textstyle#1" description,sloped},
|
||||||
|
isomorphic/.style = {ar symbol={\cong}},
|
||||||
]
|
]
|
||||||
H_1(Y, \mathbb{Z})&
|
H_1(Y, \mathbb{Z}) &
|
||||||
\times \quad H_1(Y, \mathbb{Z})&
|
\times \quad H_1(Y, \mathbb{Z})&
|
||||||
\longrightarrow &
|
\longrightarrow &
|
||||||
\quot{\mathbb{Q}}{\mathbb{Z}}
|
\quot{\mathbb{Q}}{\mathbb{Z}}
|
||||||
\text{ - a linking form}
|
\text{ - a linking form}
|
||||||
\\
|
\\
|
||||||
\quot{\mathbb{Z}^n}{A\mathbb{Z}} \ar[u,isomorphic] &
|
\quot{\mathbb{Z}^n}{A\mathbb{Z}} \ar[u,isomorphic] &
|
||||||
\quot{\mathbb{Z}^n}{A\mathbb{Z}} \ar[u,isomorphic] &\\
|
\quot{\mathbb{Z}^n}{A\mathbb{Z}} \ar[u,isomorphic] &\\
|
||||||
\end{tikzcd}
|
\end{tikzcd}
|
||||||
$(a, b) \mapsto aA^{-1}b^T$
|
$(a, b) \mapsto aA^{-1}b^T$
|
||||||
\end{center}
|
\end{center}
|
||||||
|
|
||||||
The intersection form on a four-manifold determines the linking on the boundary. \\
|
The intersection form on a four-manifold determines the linking on the boundary. \\
|
||||||
|
|
||||||
\noindent
|
\noindent
|
||||||
@ -780,4 +954,22 @@ field of fractions
|
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|
|
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\section{balagan}
|
\section{balagan}
|
||||||
|
|
||||||
|
\begin{comment}
|
||||||
|
\setlength{\arraycolsep}{2em}
|
||||||
|
\newcommand{\lbrce}{\smash{\left.\rule{0pt}{25pt}\right\}}}
|
||||||
|
\newcommand{\rbrce}{\smash{\left\{\rule{0pt}{25pt}\right.}}
|
||||||
|
\[
|
||||||
|
\begin{pmatrix}
|
||||||
|
0 & 0 & 0 \\
|
||||||
|
\sdots & \sdots\makebox[0pt][l]{$\lbrce\left\lceil\frac i2\right\rceil$} & \sdots \\
|
||||||
|
0 & 0 & \\
|
||||||
|
\hline
|
||||||
|
|
||||||
|
& & 0 \\
|
||||||
|
& & \makebox[0pt][r]{$\left\lfloor\frac i2\right\rfloor\rbrce$}\sdots \\
|
||||||
|
0 & & 0
|
||||||
|
\end{pmatrix}
|
||||||
|
\]
|
||||||
|
|
||||||
|
\end{comment}
|
||||||
\end{document}
|
\end{document}
|
||||||
|
|||||||
Loading…
Reference in New Issue
Block a user