some new text

lectures_on_knot_theory.tex

@@ -157,7 +157,7 @@ a Whitehead link:
 \includegraphics[width=0.13\textwidth]{WhiteheadLink.png},
 \item
 Borromean link:
-\includegraphics[width=0.1\textwidth]{BorromeanRings.png},
+\includegraphics[width=0.1\textwidth]{BorromeanRings.png}.
 \end{itemize}
 \end{example}
 %
@@ -214,7 +214,7 @@ Let $D$ be an oriented diagram of a link $L$. We change the diagram by smoothing
 \end{align*}
 We smooth all the crossings, so we get a disjoint union of circles on the plane. Each circle bounds a disks in $\mathbb{R}^3$ (we choose disks that don't intersect). For each smoothed crossing we add a twisted band: right-handed for a positive and left-handed for a negative one. We get an orientable surface $\Sigma$ such that $\partial \Sigma = L$.\\ 
 
-\begin{figure}[H]
+\begin{figure}[h]
 \fontsize{15}{10}\selectfont
 \centering{
 \def\svgwidth{\linewidth}
@@ -276,21 +276,21 @@ Let $\nu(\beta)$ be a tubular neighbourhood of $\beta$.  The linking number can
 \begin{example}
 \begin{itemize}
 \item
-Hopf link
+Hopf link:
 \begin{figure}[h]
 \fontsize{20}{10}\selectfont
 \centering{
 \def\svgwidth{\linewidth}
-\resizebox{0.4\textwidth}{!}{\input{images/linking_hopf.pdf_tex}}
+\resizebox{0.4\textwidth}{!}{\input{images/linking_hopf.pdf_tex}},
 }
 \end{figure}
 \item
-$T(6, 2)$ link
+$T(6, 2)$ link:
 \begin{figure}[h]
 \fontsize{20}{10}\selectfont
 \centering{
 \def\svgwidth{\linewidth}
-\resizebox{0.4\textwidth}{!}{\input{images/linking_torus_6_2.pdf_tex}}
+\resizebox{0.4\textwidth}{!}{\input{images/linking_torus_6_2.pdf_tex}}.
 }
 \end{figure}
 \end{itemize}
@@ -301,7 +301,7 @@ Let $L$ be a link and $\Sigma$ be an oriented Seifert surface for $L$. Choose a
 Let $\alpha_1^+, \dots \alpha_n^+$ be copies of $\alpha_i$ lifted up off the surface (push up along a vector field normal to $\Sigma$). Note that elements $\alpha_i$ are contained in the Seifert surface while all $\alpha_i^+$ are don't intersect the surface.
 Let $lk(\alpha_i, \alpha_j^+) = \{a_{ij}\}$. Then the matrix $S = \{a_{ij}\}_{i, j =1}^n$ is called a Seifert matrix for $L$. Note that by choosing a different basis we get a different matrix. 
 
-\begin{figure}[H]
+\begin{figure}[h]
 \fontsize{20}{10}\selectfont
 \centering{
 \def\svgwidth{\linewidth}
@@ -521,7 +521,7 @@ If $K$ is a trefoil then we can take
 $S = \begin{pmatrix}
 -1 & -1 \\
 0 & -1
-\end{pmatrix}$.
+\end{pmatrix}$. Then
 \[
 \Delta_K(t) = \det 
 \begin{pmatrix}
@@ -529,7 +529,7 @@ $S = \begin{pmatrix}
 1 & -t +1
 \end{pmatrix}
 = (t -1)^2 + t = t^2 - t +1 \ne 1
-\Rightarrow \text{trefoil is not trivial}
+\Rightarrow \text{trefoil is not trivial.}
 \]
 \end{example}
 \begin{fact}
@@ -554,7 +554,8 @@ If $\Sigma$ is a genus $g$ - Seifert surface for $K$ then $H_1(\Sigma) = \mathbb
 \end{proof}
 \begin{example}
 There are not trivial knots with Alexander polynomial equal $1$, for example: 
-$\Delta_{11n34} \equiv 1$. 
+\includegraphics[width=0.3\textwidth]{11n34.png}
+$\Delta_{11n34} \equiv 1$.
 \end{example}
 %removing one disk from surface doesn't change $H_1$ (only $H_2$)
 \section{}
@@ -582,6 +583,17 @@ A knot $K$ is smoothly slice if and only if $K$ bounds a smoothly embedded disk
 Two knots $K$ and $K^{\prime}$ are called (smoothly) concordant if there exists an annulus $A$ that is smoothly embedded in $S^3 \times [0, 1]$ such that
 $\partial A = K^{\prime} \times \{1\} \; \sqcup \; K \times \{0\} $.
 \end{definition}
+
+\begin{figure}[h]
+\fontsize{20}{10}\selectfont
+\centering{
+\def\svgwidth{\linewidth}
+\resizebox{0.8\textwidth}{!}{\input{images/concordance.pdf_tex}}
+}
+\end{figure}
+
+
+
 \noindent
 Let $m(K)$ denote a mirror image of a knot $K$. 
 \begin{fact}
@@ -666,7 +678,7 @@ Let $K \in S^1$ be a knot, $\Sigma(K)$ its double branched cover. If $V$ is a Se
 $H_1(\Sigma(K), \mathbb{Z}) \cong  \quot{\mathbb{Z}^n}{A\mathbb{Z}}$ where 
 $A = V \times V^T$, where $n = \rank V$.
 %\input{ink_diag}
-\begin{figure}[H]
+\begin{figure}[h]
 \fontsize{40}{10}\selectfont
 \centering{
 \def\svgwidth{\linewidth}
@@ -727,11 +739,11 @@ Then the intersection form can be degenerated in the sense that:
  \begin{align*}
 H_2(M, \mathbb{Z})
 \times  H_2(M, \mathbb{Z})
-\longrightarrow 
-\mathbb{Z}\\
-H_2(M, \mathbb{Z}) \longrightarrow \Hom (H_2(M, \mathbb{Z}), \mathbb{Z})\\
-(a, b) \mapsto \mathbb{Z}\\
-a \mapsto (a, \_) H_2(M, \mathbb{Z})
+&\longrightarrow 
+\mathbb{Z} \quad&
+H_2(M, \mathbb{Z}) &\longrightarrow \Hom (H_2(M, \mathbb{Z}), \mathbb{Z})\\
+(a, b) &\mapsto \mathbb{Z} \quad&
+a &\mapsto (a, \_) H_2(M, \mathbb{Z})
 \end{align*}
 has coker precisely $H_1(Y, \mathbb{Z})$. 
 \\???????????????\\
@@ -764,23 +776,27 @@ H_1(\widetilde{X}, \mathbb{Z}[t, t^{-1}]) &\longrightarrow \quot{\mathbb{Q}}{\ma
 \end{fact}
 \noindent
 Note that $\mathbb{Z}$ is not PID. Therefore we don't have primer decomposition of this moduli.  We can simplify this problem by replacing $\mathbb{Z}$ by $\mathbb{R}$. We lose some date by doing this transition. 
-\begin{flalign*}
-\xi \in S^1 \setminus \{ \pm 1\} 
+\begin{align*}
+&\xi \in S^1 \setminus \{ \pm 1\} 
 \quad
 p_{\xi} = 
-(t - \xi)(1 - \xi^{-1}) t^{-1}&\\
-\xi \in \mathbb{R} \setminus \{ \pm 1\}
+(t - \xi)(t - \xi^{-1}) t^{-1}
+\\
+&\xi \in \mathbb{R} \setminus \{ \pm 1\}
 \quad
-q_{\xi} = (t - \xi)(1 - \xi^{-1}) t^{-1}&\\
+q_{\xi} = (t - \xi)(t - \xi^{-1}) t^{-1}
+\\
+&
 \xi \notin \mathbb{R} \cup S^1 \quad
-q_{\xi} = (t - \xi)(t - \overbar{\xi})(1 - \xi^{-1})(1 - \overbar{\xi}^{-1}) t^{-2}&\\
-\Lambda = \mathbb{R}[t, t^{-1}]&\\
-\text{Then: } H_1(\widetilde{X}, \Lambda) \cong  \bigoplus_{\substack{\xi \in S^1 \setminus \{\pm 1 \}\\ k\geq 0}}
+q_{\xi} = (t - \xi)(t - \overbar{\xi})(t - \xi^{-1})(t - \overbar{\xi}^{-1}) t^{-2}\\
+&
+\Lambda = \mathbb{R}[t, t^{-1}]\\
+&\text{Then: } H_1(\widetilde{X}, \Lambda) \cong  \bigoplus_{\substack{\xi \in S^1 \setminus \{\pm 1 \}\\ k\geq 0}}
 ( \quot{\Lambda}{p_{\xi}^k })^{n_k, \xi}
 \oplus
 \bigoplus_{\substack{\xi \notin S^1 \\ l\geq 0}}
 (\quot{\Lambda}{q_{\xi}^l})^{n_l, \xi}&
-\end{flalign*}
+\end{align*}
 We can make this composition orthogonal with respect to the Blanchfield paring.
 \vspace{0.5cm}\\
 Historical remark: