matrix slice knot

lectures_on_knot_theory.tex

@@ -1067,48 +1067,11 @@ $.
 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:
-\[
-V = 
-\begin{pmatrix}
-    \begin{array}{c|c}
-           0 & *  \\
-    \hline
-           * & *
-    \end{array}
-\end{pmatrix}
-\]
-\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*}
-\newcommand\coolover[2]{\mathrlap{\smash{\overbrace{\phantom{%
-    \begin{matrix} #2 \end{matrix}}}^{\mbox{$#1$}}}}#2}
+\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]{%
@@ -1120,25 +1083,18 @@ V = \qquad
     \overbrace{XYZ}^{\mbox{$R$}}\\ \\ \\ \\ \\ \\
     \underbrace{pqr}_{\mbox{$S$}}
     \end{matrix}}%
+ V =
 \begin{matrix}% matrix for left braces
-\vphantom{a}\\
-    \coolleftbrace{A}{e \\ y\\ y}\\
-    \coolleftbrace{B}{y \\i \\ m}
+    \coolleftbrace{g_{\Sigma}}{ \\  \\ \\}
+     \\ \\ \\ \\ 
 \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}
+\begin{pmatrix}
+    \coolover{g_{\Sigma}}{0 & \dots & 0 } & * & \dots & *\\
+    \sdots &  & \sdots & \sdots &  & \sdots \\
+    0 & \dots & 0 & * & \dots & *\\
+    * & \dots & * & * & \dots & *\\
+   \sdots &  & \sdots & \sdots &  & \sdots \\
+     * & \dots & * & * & \dots & * 
+ \end{pmatrix}_{2g_{\Sigma} \times 2g_{\Sigma}}
 \end{align*}
-
-
 \end{document}