???????????????????\\
\begin{theorem}
Suppose that $K \subset S^3$ is a slice knot (i.e. $K$ bound a disk in $B^4$). 
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:
\\??????????????? T ????????
\begin{align}
PVP^{-1} = 
\begin{pmatrix}
0 & A\\
B & C
\end{pmatrix}, \quad A, C, C \in M_{g \times g} (\mathbb{Z})
\end{align}
\end{theorem}
In other words you can find rank $g$ direct summand $\mathcal{Z}$ of $H_1(F)$ \\
????????????\\
such that for any 
$\alpha, \beta \in  \mathcal{L}$ the linking number $\Lk (\alpha, \beta^+) = 0$. 
\begin{definition}
An abstract Seifert matrix (i. e. 
\end{definition}
Choose a basis $(b_1, ..., b_i)$ \\
???\\
of $H_2(Y, \mathbb{Z}$, then $A = (b_i, b_y)$ \\??\\ is a matrix of intersection form: 
\begin{align*}
\quot{\mathbb{Z}^n}{A\mathbb{Z}^n} \cong H_1(Y, \mathbb{Z}).
\end{align*}
In particular $\vert \det A\vert = \# H_1(Y, \mathbb{Z})$.\\
That means - what is happening on boundary is a measure of degeneracy.  

\begin{center}
\begin{tikzcd}
[
    column sep=tiny,
    row sep=small,
    ar symbol/.style =%
        {draw=none,"\textstyle#1" description,sloped},
         isomorphic/.style = {ar symbol={\cong}},
]
H_1(Y, \mathbb{Z}) &
  \times \quad H_1(Y, \mathbb{Z})&
  \longrightarrow &
  \quot{\mathbb{Q}}{\mathbb{Z}}
  \text{ - a linking form}
\\
   \quot{\mathbb{Z}^n}{A\mathbb{Z}} \ar[u,isomorphic] & 
   \quot{\mathbb{Z}^n}{A\mathbb{Z}} \ar[u,isomorphic] &\\
\end{tikzcd}
$(a, b) \mapsto aA^{-1}b^T$
\end{center}
?????????????????????????????????\\
\noindent
The intersection form on a four-manifold determines the linking on the boundary. \\

\noindent
\begin{fact}
Let $K \in S^1$ be a knot, $\Sigma(K)$ its double branched cover. If $V$ is a Seifert matrix for $K$, then 
\[H_1(\Sigma(K), \mathbb{Z}) \cong  \quot{\mathbb{Z}^n}{A\mathbb{Z}}\ \  ,\] where 
$A = V \times V^T$ and $n = \rank V$.
\end{fact}
%\input{ink_diag}
\begin{figure}[h]
\fontsize{20}{10}\selectfont
\centering{
\def\svgwidth{\linewidth}
\resizebox{0.5\textwidth}{!}{\input{images/ball_4.pdf_tex}}
\caption{Pushing the Seifert surface in 4-ball.}
\label{fig:pushSeifert}
}
\end{figure}
\noindent
Let $X$ be the four-manifold obtained via the double branched cover of $B^4$ branched along $\widetilde{\Sigma}$.
\begin{fact}
\begin{itemize}
\item $X$ is a smooth four-manifold, 
\item $H_1(X, \mathbb{Z}) =0$, 
\item $H_2(X, \mathbb{Z}) \cong \mathbb{Z}^n$
\item The intersection form on $X$ is $V + V^T$.
\end{itemize}
\end{fact}
\begin{figure}[h]
\fontsize{20}{10}\selectfont
\centering{
\def\svgwidth{\linewidth}
\resizebox{0.5\textwidth}{!}{\input{images/ball_4_pushed_cycle.pdf_tex}}
\caption{Cycle pushed in 4-ball.}
\label{fig:pushCycle}
}
\end{figure}
\noindent
Let  $Y = \Sigma(K)$. Then:
\begin{align*}
H_1(Y, \mathbb{Z}) \times H_1(Y, \mathbb{Z}) &\longrightarrow \quot{\mathbb{Q}}{\mathbb{Z}}
\\
(a,b) &\mapsto a A^{-1} b^{T},\qquad
A = V + V^T.
\end{align*}
????????????????????????????
\\
We have a primary decomposition of $H_1(Y, \mathbb{Z}) = U$ (as a group). For any $p \in \mathbb{P}$ we define $U_p$ to be the subgroup of elements annihilated by the same power of $p$. We have $U = \bigoplus_p U_p$. 
\begin{example}
\begin{align*}
\text{If } U &= 
\mathbb{Z}_3 \oplus
\mathbb{Z}_{45} \oplus
\mathbb{Z}_{15} \oplus
\mathbb{Z}_{75} 
\text{ then }\\
U_3 &= 
\mathbb{Z}_3 \oplus
\mathbb{Z}_9 \oplus
\mathbb{Z}_3 \oplus
\mathbb{Z}_3
\text{ and }\\
U_5 &= 
(e) \oplus
\mathbb{Z}_5 \oplus
\mathbb{Z}_5 \oplus
\mathbb{Z}_{25}.
\end{align*}
\end{example}

\begin{lemma}
Suppose $x \in U_{p_1}$, $y \in U_{p_2}$ and $p_1 \neq p_2$. Then $<x, y > = 0$.
\end{lemma}
\begin{proof}
\begin{align*}
x \in U_{p_1} 
\end{align*}
\end{proof}
\begin{align*}
H_1(Y, \mathbb{Z}) \cong \quot{\mathbb{Z}^n}{A\mathbb{Z}}\\
A \longrightarrow BAC^T \quad \text{Smith normal form}
\end{align*}
???????????????????????\\
In general 

%no lecture at 29.04