lectures_on_knot_theory/lec_25_03.tex

\subsection{Slice knots and metabolic form}
\begin{theorem}
\label{the:sign_slice}
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}
If $V$ is a Hermitian matrix ($\overline{V} = V^T$) of size  $2n \times 2n$, 
$
V = \begin{pmatrix}
0 & A \\
\overline{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
$\begin{pmatrix}
0 & A\\
\overline{A^T} & B
\end{pmatrix}$ with half-dimensional null-space. 
\end{definition}
\noindent
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:
\begin{align*}
\det((1 - t) S + (1 - \bar{t}) S^T) =& 
\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}
If $K \sim K^\prime$ then for all but finitely many $t \in S^1 \setminus \{1\}: \sigma_K(t) = -\sigma_{K^\prime}(t)$.
\end{corollary}
\begin{proof}
If $ K  \sim K^\prime$ then $K \# K^\prime$ is slice. 
\[
\sigma_{-K^\prime}(t) = -\sigma_{K^\prime}(t)
\]
The signature gives a homomorphism from the concordance group to $\mathbb{Z}$.
Remark: if $t \in S^1$ is not algebraic over $\mathbb{Z}$, then $\sigma_K(t) \neq 0$
(we can use the argument that $\mathscr{C} \longrightarrow \mathbb{Z}$ as well). 
\end{proof}
\subsection{Four genus}
\begin{figure}[h]
\fontsize{20}{10}\selectfont
\centering{
\def\svgwidth{\linewidth}
\resizebox{0.7\textwidth}{!}{\input{images/genus_2_bordism.pdf_tex}}
}
\caption{$K$ and $K^\prime$ are connected by a genus $g$ surface.}\label{fig:genus_2_bordism}
\end{figure}

\begin{proposition}[Kawauchi inequality]
If there exists a genus $g$ surface as in Figure \ref{fig:genus_2_bordism}
then for almost all 
$t \in S^1 \setminus \{1\}$ we have 
$\vert 
\sigma_K(t) - \sigma_{K^\prime}(t) 
\vert \leq 2 g$. 
\end{proposition}
% Kawauchi  Chapter 12 ???
% Borodzik 2010 Morse theory for plane algebraic curves
\begin{lemma}
If $K$ bounds a genus $g$ surface $X \in B^4$ and $S$ is a Seifert form then ${S \in M_{2n \times 2n}}$ has a block structure $\begin{pmatrix}
0 & A\\
B & C
\end{pmatrix}$, where $0$ is $(n - g) \times (n - g)$ submatrix.
\end{lemma}

\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}
\noindent
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}
\begin{corollary}
If $t$ is not a root of 
$\det (tS - S^T) $, then
$\vert \sigma_K(t) \vert \leq 2g$.
\end{corollary}
\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}

\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
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}
\begin{example}
\begin{itemize}
\item Let $K = T(2, 3)$. $\sigma(K) = -2$, therefore $T(2, 3)$ isn't a slice knot. 
\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,
\item
the equality: 
\[
g_4(T(p, q) ) = \frac{1}{2} (p - 1) (g -1)
\]
was conjecture in the '70 and proved by  P. Kronheimer and T. Mrówka (1994).
%  OZSVATH-SZABO AND RASMUSSEN
\end{itemize}
\end{example}
\begin{proposition}
$g_4 (T(p, q) \# -T(r, s))$ is in general hopelessly unknown. 
\end{proposition}
\begin{proposition}
Supremum of the signature function of the knot is bounded almost everywhere by two times $4$ - genus:
\[
\ess \sup \vert \sigma_K(t) \vert \leq 2 g_4(K).
\]
\end{proposition}
\subsection{Topological genus}
\begin{definition}
A knot $K$ is called topologically slice if $K$ bounds a topological locally flat disc in $B^4$ (i.e. the disk has tubular neighbourhood). 
\end{definition}
\begin{theorem}[Freedman, '82]
If $\Delta_K(t) = 1$, then $K$ is topologically slice (but not necessarily smoothly slice).  
\end{theorem}
\begin{theorem}[Powell, 2015]
If $K$ is genus $g$ 
(topologically flat) 
cobordant to $K^\prime$, 
then 
\[
\vert \sigma_K(t)  - \sigma_{K^\prime}(t) \vert \leq 2 g
\]
if $g_4^{\mytop}(K) \geq \ess \sup \vert \sigma_K(t) \vert$.
\end{theorem}
\noindent
The proof for smooth category was based on following equality:
\[
\dim \ker (H_1 (Y) \longrightarrow H_1(\Omega)) = \frac{1}{2} \dim H_1(Y).
\]
For this equality we assumed that there exists a $3$ - dimensional manifold $\Omega$ (as shown in Figure \ref{fig:omega_in_B_4}) which was guaranteed by Pontryagin-Thom Construction.\\
Pontryagin-Thom Construction relays on taking $\Omega$ as preimage of regular value:
\[
H^1 (B^4 \setminus Y, \mathbb{Z}) = [B^4 \setminus Y, S^1], 
\]
what relies on Sard's theorem, that the set of regular values has positive measure. But Sard's theorem doesn't work for topologically locally flat category. So there was a gap in the proof for topological locally flat category - the existence of $\Omega$.\\
\noindent
Remark: unless $p=2$ or $p = 3 \wedge  q = 4$:
\[
g_4^{\mytop} (T(p, q)) < q_4(T(p, q)).
\]
% Wilczyński '93 
%Feller 2014
%Baoder 2017
%Lemark
\\
\noindent
From the category of cobordant knots (or topologically cobordant knots) there exists a map to $\mathbb{Z}$ given by signature function. To any element $K$ we can associate a form 
\[
(1 - t)S + (1 - \bar{t})S^T) \in W(\mathbb{Z}[t, t^{-1}]).
\] This association is not well define because id depends on the choice of Seifert form. However, different choices lead ever to congruent forms ($S \mapsto CSC^T$) or induced the change on the form by adding or subtracting a hyperbolic element. 
\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
A form is meant as hermitian with respect to this involution: $A^T = A: (a, b) = \bar{(a, b)}$.
\\
$
W(\mathbb{Z}_p) = \mathbb{Z}_2 \oplus 
\mathbb{Z}_2$ or 
$\mathbb{Z}_4$
\\
???????????????????????
\\
$\sum a_gt^j \longrightarrow \sum a_g t^{-1}$\\
\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}