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\subsection { Slice knots and metabolic form}
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\begin { theorem}
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\label { the:sign_ slice}
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If $ K $ is slice,
then $ \sigma _ K ( t )
= \sign ( (1 - t)S +(1 - \bar { t} )S^ T)$
is zero except possibly of finitely many points and $ \sigma _ K ( - 1 ) = \sign ( S + S ^ T ) \neq 0 $ .
\end { theorem}
\begin { lemma}
\label { lem:metabolic}
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If $ V $ is a Hermitian matrix ($ \bar { V } = V ^ T $ ), $ V $ is of size $ 2 n \times 2 n $ ,
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$
V = \begin { pmatrix}
0 & A \\
\bar { A} ^ T & B
\end { pmatrix}
$ and $ \det V \neq 0$ then $ \sigma (V) = 0$ .
\end { lemma}
\begin { definition}
A Hermitian form $ V $ is metabolic if $ V $ has structure
$ \begin { pmatrix }
0 & A\\
\bar { A} ^ T & B
\end { pmatrix} $ with half - dimensional null - space.
\end { definition}
\noindent
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Theorem \ref { the:sign_ slice} can be also express as follow:
non-degenerate metabolic hermitian form has vanishing signature.
\begin { proof}
\noindent
We note that $ \det ( S + S ^ T ) \neq 0 $ . Hence $ \det ( ( 1 - t ) S + ( 1 - \bar { t } ) S ^ T ) $ is not identically zero on $ S ^ 1 $ , so it is non-zero except possibly at finitely many points. We apply the Lemma \ref { lem:metabolic} .
\\
Let $ t \in S ^ 1 \setminus \{ 1 \} $ .
Then:
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\begin { align*}
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\det ((1 - t) S + (1 - \bar { t} ) S^ T) =&
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\det ((1 - t) S + (t\bar { t} - \bar { t} ) S^ T) =\\
& \det ((1 - t) (S - \bar { t} - S^ T)) =
\det ((1 -t)(S - \bar { t} S^ T)).
\end { align*}
As $ \det ( S + S ^ T ) \neq 0 $ , so $ S - \bar { t } S ^ T \neq 0 $ .
\end { proof}
\begin { corollary}
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If $ K \sim K ^ \prime $ then for all but finitely many $ t \in S ^ 1 \setminus \{ 1 \} : \sigma _ K ( t ) = - \sigma _ { K ^ \prime } ( t ) $ .
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\end { corollary}
\begin { proof}
If $ K \sim K ^ \prime $ then $ K \# K ^ \prime $ is slice.
\[
\sigma _ { -K^ \prime } (t) = -\sigma _ { K^ \prime } (t)
\]
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The signature gives a homomorphism from the concordance group to $ \mathbb { Z } $ .
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Remark: if $ t \in S ^ 1 $ is not algebraic over $ \mathbb { Z } $ , then $ \sigma _ K ( t ) \neq 0 $
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(we can use the argument that $ \mathscr { C } \longrightarrow \mathbb { Z } $ as well).
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\end { proof}
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\subsection { Four genus}
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\begin { figure} [h]
\fontsize { 20} { 10} \selectfont
\centering {
\def \svgwidth { \linewidth }
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\resizebox { 0.7\textwidth } { !} { \input { images/genus_ 2_ bordism.pdf_ tex} }
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}
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\caption { $ K $ and $ K ^ \prime $ are connected by a genus $ g $ surface.} \label { fig:genus_ 2_ bordism}
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\end { figure}
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\begin { proposition} [Kawauchi inequality]
If there exists a genus $ g $ surface as in Figure \ref { fig:genus_ 2_ bordism}
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then for almost all
$ t \in S ^ 1 \setminus \{ 1 \} $ we have
$ \vert
\sigma _ K(t) - \sigma _ { K^ \prime } (t)
\vert \leq 2 g$ .
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\end { proposition}
% Kawauchi Chapter 12 ???
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% Borodzik 2010 Morse theory for plane algebraic curves
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\begin { lemma}
If $ K $ bounds a genus $ g $ surface $ X \in B ^ 4 $ and $ S $ is a Seifert form then $ { S \in M _ { 2 n \times 2 n } } $ has a block structure $ \begin { pmatrix }
0 & A\\
B & C
\end { pmatrix} $ , where $ 0$ is $ (n - g) \times (n - g)$ submatrix.
\end { lemma}
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\begin { proof}
\begin { figure} [h]
\fontsize { 20} { 10} \selectfont
\centering {
\def \svgwidth { \linewidth }
\resizebox { 0.5\textwidth } { !} { \input { images/genus_ bordism_ zeros.pdf_ tex} }
}
\caption { There exists a $ 3 $ - manifold $ \Omega $ such that $ \partial \Omega = X \cup \Sigma $ .} \label { fig:omega_ in_ B_ 4}
\end { figure}
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\noindent
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Let $ K $ be a knot and $ \Sigma $ its Seifert surface as in Figure \ref { fig:omega_ in_ B_ 4} .
There exists a $ 3 $ - submanifold
$ \Omega $ such that
$ \partial \Omega = Y = X \cup \Sigma $
(by Thom-Pontryagin construction).
If $ \alpha , \beta \in \ker ( H _ 1 ( \Sigma ) \longrightarrow H _ 1 ( \Omega ) ) $ ,
then $ { \Lk ( \alpha , \beta ^ + ) = 0 } $ . Now we have to determine the size of the kernel. We know that
$ { \dim H _ 1 ( \Sigma ) = 2 n } $ . When we glue $ \Sigma $ (genus $ n $ ) and $ X $ (genus $ g $ ) along a circle we get a surface of genus $ n + g $ . Therefore $ \dim H _ 1 ( Y ) = 2 n + 2 g $ . Then:
\[
\dim (\ker (H_ 1(Y) \longrightarrow H_ 1(\Omega )) = n + g.
\]
So we have $ H _ 1 ( W ) $ of dimension
$ 2 n + 2 g $
- the image of $ H _ 1 ( Y ) $
with a subspace
corresponding to the image of $ H _ 1 ( \Sigma ) $ with dimension $ 2 n $ and a subspace corresponding to the kernel
of $ H _ 1 ( Y ) \longrightarrow H _ 1 ( \Omega ) $ of size $ n + g $ .
We consider minimal possible intersection of this subspaces that corresponds to the kernel of the composition $ H _ 1 ( \Sigma ) \longrightarrow H _ 1 ( Y ) \longrightarrow H _ 1 ( \Omega ) $ . As the first map is injective, elements of the kernel of the composition have to be in the kernel of the second map.
So we can calculate:
\[
\dim \ker (H_ 1(\Sigma ) \longrightarrow H_ 1(\Omega )) = 2 n + n + g -2 n - 2 g = n - g.
\]
\end { proof}
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\begin { corollary}
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If $ t $ is not a root of
$ \det S S ^ T - $ \\
????????????????\\
then
$ \vert \sigma _ K ( t ) \vert \leq 2 g $ .
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\end { corollary}
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\begin { fact}
If there exists cobordism of genus $ g $ between $ K $ and $ K ^ \prime $ like shown in Figure \ref { fig:proof_ for_ bound_ disk} , then $ K \# - K ^ \prime $ bounds a surface of genus $ g $ in $ B ^ 4 $ .
\end { fact}
\begin { figure} [H]
\fontsize { 20} { 10} \selectfont
\centering {
\def \svgwidth { \linewidth }
\resizebox { 0.7\textwidth } { !} { \input { images/genus_ bordism_ proof.pdf_ tex} }
}
\caption { If $ K $ and $ K ^ \prime $ are connected by a genus $ g $ surface, then $ K \# - K ^ \prime $ bounds a genus $ g $ surface.} \label { fig:proof_ for_ bound_ disk}
\end { figure}
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\begin { definition}
The (smooth) four genus $ g _ 4 ( K ) $ is the minimal genus of the surface $ \Sigma \in B ^ 4 $ such that $ \Sigma $ is compact, orientable and $ \partial \Sigma = K $ .
\end { definition}
\noindent
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Remarks:
\begin { enumerate} [label={ (\arabic * )} ]
\item
$ 3 $ - genus is additive under taking connected sum, but $ 4 $ - genus is not,
\item
for any knot $ K $ we have $ g _ 4 ( K ) \leq g _ 3 ( K ) $ .
\end { enumerate}
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\begin { example}
\begin { itemize}
\item Let $ K = T ( 2 , 3 ) $ . $ \sigma ( K ) = - 2 $ , therefore $ T ( 2 , 3 ) $ isn't a slice knot.
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\item Let $ K $ be a trefoil and $ K ^ \prime $ a mirror of a trefoil. $ g _ 4 ( K ^ \prime ) = 1 $ , but $ g _ 4 ( K \# K ^ \prime ) = 0 $ , so we see that $ 4 $ -genus isn't additive,
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\item
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the equality:
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\[
g_ 4(T(p, q) ) = \frac { 1} { 2} (p - 1) (g -1)
\]
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was conjecture in the '70 and proved by P. Kronheimer and T. Mrówka (1994).
% OZSVATH-SZABO AND RASMUSSEN
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\end { itemize}
\end { example}
\begin { proposition}
$ g _ 4 ( T ( p, q ) \# - T ( r, s ) ) $ is in general hopelessly unknown.
\\ ???????????????\\
essentially $ \sup \vert \sigma _ K ( t ) \vert \leq 2 g _ n ( K ) $
\end { proposition}
\begin { definition}
A knot $ K $ is called topologically slice if $ K $ bounds a topological locally flat disc in $ B ^ 4 $ (it has tubular neighbourhood).
\end { definition}
\begin { theorem} [Freedman, '82]
If $ \Delta _ K ( t ) \geq 1 $ , then $ K $ is topologically slice, but not necessarily smoothly slice.
\end { theorem}
\begin { theorem} [Powell, 2015]
If $ K $ is genus g
\\ (top. loc.?????????)\\
cobordant to $ K ^ \prime $ ,
then $ \vert \sigma _ K ( t ) - \sigma _ { K ^ \prime } ( t ) \vert \leq 2 g $ . \\
If $ g _ 4 ^ { \mytop } ( K ) \geq $ ?????ess $ \sup \vert \sigma _ K ( t ) \vert $ and ?????????\\
$ \dim \ker ( H _ 1 ( Y ) \longrightarrow H _ 1 ( \Omega ) ) = \frac { 1 } { 2 } \dim H _ 1 ( Y ) $ .
\end { theorem}
???????????????
\[
H^ 1 (B^ 4 \setminus Y, \mathbb { Z} ) = [B^ 4 \setminus Y, S^ 1]
\]
\noindent
Remark: unless $ p = 2 $ or $ p = 3 \wedge q = 4 $ :
\[
g_ 4^ \top (T(p, q)) < q_ 4(T(p, q))
\]
%??????????????????????
\begin { definition}
The Witt group $ W $ of $ \mathbb { Z } [ t, t ^ { - 1 } ] $ elements are classes of non-degenerate
forms over $ \mathbb { Z } [ t, t ^ { - 1 } ] $ under the equivalence relation $ V \sim W $ if $ V \oplus - W $ is metabolic.
\end { definition}
\noindent
If $ S $ differs from $ S ^ \prime $ by a row extension, then
$ ( 1 - t ) S + ( 1 - \bar { t } ^ { - 1 } ) S ^ T $ is Witt equivalence to $ ( 1 - t ) S ^ \prime + ( 1 - t ^ { - 1 } ) S ^ T $ .
%???????????????????????????
\noindent
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A form is meant as hermitian with respect to this involution: $ A ^ T = A: ( a, b ) = \bar { ( a, b ) } $ .
\\
????????????????????????????
\\
\begin { theorem} [Levine '68]
\[
W(\mathbb { Z} [t^ { \pm 1} )
\longrightarrow \mathbb { Z} _ 2^ \infty \oplus
\mathbb { Z} _ 4^ \infty \oplus
\mathbb { Z}
\]
\end { theorem}