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\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:
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\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 ) $ \\
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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 ) $ \\
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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}
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\noindent
The intersection form on a four-manifold determines the linking on the boundary. \\
\noindent
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\begin { fact}
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Let $ K \in S ^ 1 $ be a knot, $ \Sigma ( K ) $ its double branched cover. If $ V $ is a Seifert matrix for $ K $ , then
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\[ H _ 1 ( \Sigma ( K ) , \mathbb { Z } ) \cong \quot { \mathbb { Z } ^ n } { A \mathbb { Z } } \ \ , \] where
$ A = V \times V ^ T $ and $ n = \rank V $ .
\end { fact}
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%\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*}
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\\
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*}
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In general
%no lecture at 29.04