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matrix slice knot

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Maria Marchwicka 2019-06-08 14:14:33 +02:00
parent d2146a556c
commit 401a9233b9
1 changed files with 14 additions and 58 deletions
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lectures_on_knot_theory.tex
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@ -1067,47 +1067,10 @@ $.
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\\ 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: has a subspace of dimension $g_{\Sigma}$ on which it is zero:
\[
V =
\begin{pmatrix}
\begin{array}{c|c}
0 & * \\
\hline
* & *
\end{array}
\end{pmatrix}
\]
\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
$}%
\kern0pt
}%
}%
}
V = \qquad
\begin{pmatrix}
0 & \cdots & 0 & * & \cdots & * \\
\vdots & & \vdots & \vdots & &\vdots \\
0 & \cdots & 0 & * & \cdots & *
\AddLeft{g_{\Sigma}}\\
* & \cdots & * & * & \cdots & * \\
\vdots & & \vdots & \vdots & &\vdots \\
* & \cdots & * & * & \cdots & *
\end{pmatrix}_{2g_{\Sigma} \times 2g_{\Sigma}}
\end{align*}
\begin{align*} \begin{align*}
\newcommand\coolover[2]{\mathrlap{\smash{\overbrace{\phantom{% \newcommand\coolover[2]%
{\mathrlap{\smash{\overbrace{\phantom{%
\begin{matrix} #2 \end{matrix}}}^{\mbox{$#1$}}}}#2} \begin{matrix} #2 \end{matrix}}}^{\mbox{$#1$}}}}#2}
\newcommand\coolunder[2]{\mathrlap{\smash{\underbrace{\phantom{% \newcommand\coolunder[2]{\mathrlap{\smash{\underbrace{\phantom{%
\begin{matrix} #2 \end{matrix}}}_{\mbox{$#1$}}}}#2} \begin{matrix} #2 \end{matrix}}}_{\mbox{$#1$}}}}#2}
@ -1120,25 +1083,18 @@ V = \qquad
\overbrace{XYZ}^{\mbox{$R$}}\\ \\ \\ \\ \\ \\ \overbrace{XYZ}^{\mbox{$R$}}\\ \\ \\ \\ \\ \\
\underbrace{pqr}_{\mbox{$S$}} \underbrace{pqr}_{\mbox{$S$}}
\end{matrix}}% \end{matrix}}%
V =
\begin{matrix}% matrix for left braces \begin{matrix}% matrix for left braces
\vphantom{a}\\ \coolleftbrace{g_{\Sigma}}{ \\ \\ \\}
\coolleftbrace{A}{e \\ y\\ y}\\ \\ \\ \\ \\
\coolleftbrace{B}{y \\i \\ m}
\end{matrix}% \end{matrix}%
\begin{bmatrix} \begin{pmatrix}
a & \coolover{R}{b & c & d} & x & \coolover{Z}{x & x}\\ \coolover{g_{\Sigma}}{0 & \dots & 0 } & * & \dots & *\\
e & f & g & h & x & x & x \\ \sdots & & \sdots & \sdots & & \sdots \\
y & y & y & y & y & y & y \\ 0 & \dots & 0 & * & \dots & *\\
y & y & y & y & y & y & y \\ * & \dots & * & * & \dots & *\\
y & y & y & y & y & y & y \\ \sdots & & \sdots & \sdots & & \sdots \\
i & j & k & l & x & x & x \\ * & \dots & * & * & \dots & *
m & \coolunder{S}{n & o} & \coolunder{W}{p & x & x} & x \end{pmatrix}_{2g_{\Sigma} \times 2g_{\Sigma}}
\end{bmatrix}%
\begin{matrix}% matrix for right braces
\coolrightbrace{x \\ x \\ y\\ y}{T}\\
\coolrightbrace{y \\ y \\ x }{U}
\end{matrix}
\end{align*} \end{align*}
\end{document} \end{document}