macros

lectures_on_knot_theory.tex

@@ -1047,6 +1047,37 @@ A square hermitian matrix $A$ of size $n$.
 
 field of fractions
 
+\section{\hfill\DTMdate{2019-06-03}}
+\begin{theorem}
+Let $K$ be a knot and $u(K)$ its unknotting number. Let $g_4(K)$ be a minimal four genus of a smooth surface $S$ in $B^4$ such that $\partial S = K$. Then:
+\[
+u(K) \geq g_4(K)
+\]
+\begin{proof}
+Recall that if $u(K)=u$ then $K$ bounds a disk $\Delta$ with $u$ ordinary double points. 
+\\
+\noindent
+Remove from $\Delta$ the two self intersecting and glue the Seifert surface for the Hopf link. The reality surface $S$ has Euler characteristic $\chi(S) = 1 - 2u$. Therefore $g_4(S) = u$ .
+\end{proof}
+???????????????????\\
+\begin{example}
+The knot $8_{20}$ is slice: $\sigma \equiv 0$ almost everywhere but $\sigma(e^{\frac{ 2\pi i}{6}}) = + 1$.
+\end{example}
+\subsection{Surgery}
+Recall that $H_1(S^1 \times S^1, \mathbb{Z}) = \mathbb{Z}^3$. As generators for $H_1$ we can set ${\alpha = [S^1 \times \{pt\}]}$ and ${\beta=[\{pt\} \times S^1]}$. Suppose ${\phi: S^1 \times S^1 \longrightarrow S^1 \times S^1}$ is a diffeomorphism. 
+Consider an induced map on homology group: 
+\begin{align*}
+H_1(S^1 \times S^1, \mathbb{Z}) \ni \phi_* (\alpha) &= p\alpha + q \beta, \quad p, q \in \mathbb{Z},\\
+\phi_*(\beta) &= r \alpha + s \beta, \quad r, s \in \mathbb{Z}, \\
+\phi_* &= 
+  \begin{pmatrix}
+     p & q\\
+     r & s
+  \end{pmatrix}
+\end{align*} 
+
+\end{theorem}
+
 
 \section{balagan}
 
@@ -1064,7 +1095,7 @@ Therefore $\dim_{\mathbb{Q}} \quot{H_1(Y)}{V}
 $.
 \end{proof}
 \noindent
-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\\
+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 a $4$ - manifolds???\\
 ?????\\
 has a subspace of dimension $g_{\Sigma}$ on which it is zero: