1, 2, 4 lec

lec_2.tex

@@ -208,7 +208,7 @@ $\Delta_{11n34} \equiv 1$.
 We know that $3$ - sphere can be obtained by gluing two solid tori:
 $S^3 = \partial D^4 = \partial (D^2 \times D^2) =  (D^2 \times S^1) \cup (S^1 \times D^2)$. So the complement of solid torus in $S^3$ is another solid torus.\\
 Analytically it can be describes as follow. 
-Take $(z_1, z_2) \in \mathbb{C}$ such  that $max(\mid z_1 \mid, \mid z_2\mid) = 1
+Take $(z_1, z_2) \in \mathbb{C}$ such  that $\max(\mid z_1 \mid, \mid z_2\mid) = 1
 $. Define following sets: $S_1 = \{ (z_1, z_2) \in S^3: \mid z_1 \mid = 0\} \cong S^1 \times D^2 $ and $S_2 = \{(z_1, z_2) \in S ^3: \mid z_2 \mid = 1 \} \cong  D^2 \times S^1$. The intersection $S_1 \cap S_2 = \{(z_1, z_2): \mid z_1 \mid = \mid z_2 \mid = 1 \} \cong S^1 \times S^1$
 \begin{figure}[h]
 \centering{

lectures_on_knot_theory.tex

@@ -89,6 +89,8 @@
 \DeclareMathOperator{\Sl}{SL}
 \DeclareMathOperator{\Lk}{lk}
 \DeclareMathOperator{\pt}{\{pt\}}
+\DeclareMathOperator{\sign}{sign}
+
 
 \titleformat{\subsection}{%
      \normalfont \fontsize{12}{15}\bfseries}{%
@@ -141,9 +143,26 @@
 \end{figure}
 ????????????
 \\
+$L$ is a link in $S^3$ \\
+\\?????????????????\\
+$L$ is an unknot if and only if $F(0) \neq 0$ (provided $S^3$ has a sufficiently small radius.
+\\
 \noindent
+Remark: if $S^3$ is large it can happen that $L$ is unlink, but $F^{-1} \cap B^4$ is "complicated". \\
+????????????\\
+\noindent
+\begin{example}
+Let $p$ and $q$ be coprime numbers such that $p<q$ and $p,q>1$.
+\\
+$F^{-1}(0) \cap S^3$ is a solid torus $T(p, q)$. \\
+$F(z, w) = z^p - w^q$\\
+Consider $S^3 = \{ (z, w) \in \mathbb{C} : \max( \mid z \mid, \mid w \mid ) = \varepsilon$.\\
+$F^{-1}(0) = \{t = t^q, w = t^p\}.$ For unknot $t = \max (\mid t\mid ^p, \mid t \mid^q) = \varepsilon$.
+\end{example}
 as a corollary we see that $K_T^{n, }$ ???? \\
-is not slice unless $m=0$. 
+is not slice unless $m=0$. \\
+$t = re^{i \Theta}, \Theta \in [0, 2\pi], r = \varepsilon^{\frac{i}{p}}$ ?????????????????????????\\
+Suppose $L$ is a diagonal link. $L = F^{-1}(0) \cap S^3$.
 \begin{theorem}
 The map $j: \mathscr{C} \longrightarrow \mathbb{Z}^{\infty}$ is a surjection that maps ${K_n}$ to a linear independent set. Moreover $\mathscr{C} \cong \mathbb{Z}$
 \end{theorem}
@@ -178,6 +197,23 @@ A link $L$ is fibered if there exists a map ${\phi: S^3\setminus L \longleftarro
 
 
 \section{\hfill\DTMdate{2019-03-25}}
+\begin{theorem}
+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{proof}
+\begin{lemma}
+If $V$ is a Hermitian matrix ($\bar{V} = V^T$), $V$ is of size  $2n \times 2n$ and 
+$
+V = \begin{pmatrix}
+0 & A \\
+\bar{A}^T & B
+\end{pmatrix}
+$
+\end{lemma}
+\end{proof}
 \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}
@@ -198,7 +234,15 @@ H_2(X, \mathbb{Z}) \longrightarrow \mathbb{Z}$ - symmetric, non singular.
 \\
 Let $A$ and $B$ be closed, oriented surfaces in $X$. 
 \begin{proposition}
-$A \cdot B$ doesn't depend of choice of $A$ and $B$ in their homology classes. 
+$A \cdot B$ doesn't depend of choice of $A$ and $B$ in their homology classes $[A], [B] \in H_2(X, \mathbb{Z})$. \\
+\noindent
+If $M$ is an $m$ - dimensional close, connected and orientable manifold, then $H_m(M, \mathbb{Z})$ and the orientation if $M$ determined a cycle $[M] \in H_m(M, \mathbb{Z})$, called the fundamental cycle. 
+\begin{example}
+If $\omega$ is an $m$ - form then: 
+\[
+ = [\omega]
+\]
+\end{example} 
 %$A \cdot B$  gives the pairing as ??
  \end{proposition}