delete unused

images/BorromeanRings.svg

@@ -0,0 +1,719 @@
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+</svg>

images/ball_4.pdf_tex

@@ -36,7 +36,7 @@
   }%
   \providecommand\rotatebox[2]{#2}%
   \ifx\svgwidth\undefined%
-    \setlength{\unitlength}{359.36692366bp}%
+    \setlength{\unitlength}{640.00045656bp}%
     \ifx\svgscale\undefined%
       \relax%
     \else%
@@ -48,16 +48,17 @@
   \global\let\svgwidth\undefined%
   \global\let\svgscale\undefined%
   \makeatother%
-  \begin{picture}(1,1.21771409)%
+  \begin{picture}(1,0.33301347)%
     \put(0,0){\includegraphics[width=\unitlength,page=1]{ball_4.pdf}}%
-    \put(0.18154802,0.31666994){\color[rgb]{0,0,0}\makebox(0,0)[lt]{\begin{minipage}{3.07104704\unitlength}\raggedright $B^4$ \end{minipage}}}%
+    \put(0.09657807,0.16873207){\color[rgb]{0,0,0}\makebox(0,0)[lt]{\begin{minipage}{1.7204386\unitlength}\raggedright $B^4$\end{minipage}}}%
     \put(0,0){\includegraphics[width=\unitlength,page=2]{ball_4.pdf}}%
-    \put(0.18066707,0.93882205){\color[rgb]{0,0,0}\makebox(0,0)[lt]{\begin{minipage}{3.06394788\unitlength}\raggedright $B^4$\end{minipage}}}%
+    \put(0.43862137,0.2802709){\color[rgb]{0,0,0}\makebox(0,0)[lt]{\begin{minipage}{0.15019764\unitlength}\raggedright \end{minipage}}}%
+    \put(0.35652029,0.19426923){\color[rgb]{0,0,0}\makebox(0,0)[lt]{\begin{minipage}{0.17633501\unitlength}\raggedright $\Sigma$\end{minipage}}}%
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images/seifert_alg.pdf_tex

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images/torus_1_2_3.pdf_tex

@@ -36,7 +36,7 @@
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+    \put(0.19333753,0.06190796){\color[rgb]{0,0,0}\makebox(0,0)[lt]{\begin{minipage}{0.22186975\unitlength}\raggedright \shortstack{genus $0$}\\  \end{minipage}}}%
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+    \put(0.85136435,0.24672362){\color[rgb]{0,0,0}\makebox(0,0)[lt]{\begin{minipage}{0.13890231\unitlength}\raggedright genus $2$\\  \end{minipage}}}%
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images/torus_1_2_3_v2.pdf_tex

@@ -1,66 +0,0 @@
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lectures_on_knot_theory.tex

@@ -227,9 +227,9 @@ We smooth all the crossings, so we get a disjoint union of circles on the plane.
 \noindent
 Note: in general the obtained surface doesn't need to be connected, but by taking connected sum of all components we can easily get a connected surface (i.e. we take two disconnected components and cut a disk in each of them: $D_1$ and $D_2$; now we glue both components on the boundaries: $\partial D_1$ and $\partial D_2$. 
 
-\begin{figure}[H]
+\begin{figure}[h]
 \begin{center}
-\includegraphics[width=0.4\textwidth]{seifert_connect.png}
+\includegraphics[width=0.6\textwidth]{seifert_connect.png}
 \end{center}
 \caption{Connecting two surfaces.}
 \label{fig:SeifertConnect}
@@ -239,11 +239,11 @@ Note: in general the obtained surface doesn't need to be connected, but by takin
 Every link in $S^3$ bounds a surface $\Sigma$ that is compact, connected and orientable. Such a surface is called a Seifert surface.
 \end{theorem}
 %
-\begin{figure}[H]
+\begin{figure}[h]
-\fontsize{15}{10}\selectfont
+\fontsize{12}{10}\selectfont
 \centering{
 \def\svgwidth{\linewidth}
-\resizebox{0.8\textwidth}{!}{\input{images/torus_1_2_3.pdf_tex}}
+\resizebox{1\textwidth}{!}{\input{images/torus_1_2_3.pdf_tex}}
 \caption{Genus of an orientable surface.}
 \label{fig:genera}
 }
@@ -277,7 +277,7 @@ Let $\nu(\beta)$ be a tubular neighbourhood of $\beta$.  The linking number can
 \begin{itemize}
 \item
 Hopf link
-\begin{figure}[H]
+\begin{figure}[h]
 \fontsize{20}{10}\selectfont
 \centering{
 \def\svgwidth{\linewidth}
@@ -286,7 +286,7 @@ Hopf link
 \end{figure}
 \item
 $T(6, 2)$ link
-\begin{figure}[H]
+\begin{figure}[h]
 \fontsize{20}{10}\selectfont
 \centering{
 \def\svgwidth{\linewidth}
@@ -686,12 +686,15 @@ Let $X$ be the four-manifold obtained via the double branched cover of $B^4$ bra
 \end{fact}
 \noindent
 Let  $Y = \Sigma(K)$. Then:
-\begin{align*}
+\begin{flalign*}
-&H_1(Y, \mathbb{Z}) \times H_1(Y, \mathbb{Z}) \longrightarrow \quot{\mathbb{Q}}{\mathbb{Z}}\\ &(a,b) \mapsto a A^{-1} b^{T},\qquad
+H_1(Y, \mathbb{Z}) \times H_1(Y, \mathbb{Z}) \longrightarrow \quot{\mathbb{Q}}{\mathbb{Z}}&
-A = V + V^T\\
+\\
-&H_1(Y, \mathbb{Z}) \cong \quot{\mathbb{Z}^n}{A\mathbb{Z}}\\
+(a,b) \mapsto a A^{-1} b^{T},\qquad
-&A \longrightarrow BAC^T \quad \text{Smith normal form}
+A = V + V^T&
-\end{align*}
+\\
+H_1(Y, \mathbb{Z}) \cong \quot{\mathbb{Z}^n}{A\mathbb{Z}}&\\
+A \longrightarrow BAC^T \quad \text{Smith normal form}&
+\end{flalign*}
 ???????????????????????\\
 In general 
 
@@ -762,23 +765,23 @@ 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{align*}
+\begin{flalign*}
-&\xi \in S^1 \setminus \{ \pm 1\} 
+\xi \in S^1 \setminus \{ \pm 1\} 
 \quad
 p_{\xi} = 
-(t - \xi)(1 - \xi^{-1}) t^{-1}\\
+(t - \xi)(1 - \xi^{-1}) t^{-1}&\\
-&\xi \in \mathbb{R} \setminus \{ \pm 1\}
+\xi \in \mathbb{R} \setminus \{ \pm 1\}
 \quad
-q_{\xi} = (t - \xi)(1 - \xi^{-1}) t^{-1}\\
+q_{\xi} = (t - \xi)(1 - \xi^{-1}) t^{-1}&\\
-&\xi \notin \mathbb{R} \cup S^1 \quad
+\xi \notin \mathbb{R} \cup S^1 \quad
-q_{\xi} = (t - \xi)(t - \overbar{\xi})(1 - \xi^{-1})(1 - \overbar{\xi}^{-1}) t^{-2}\\
+q_{\xi} = (t - \xi)(t - \overbar{\xi})(1 - \xi^{-1})(1 - \overbar{\xi}^{-1}) t^{-2}&\\
-&\Lambda = \mathbb{R}[t, t^{-1}]\\
+\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}}
+\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}
+(\quot{\Lambda}{q_{\xi}^l})^{n_l, \xi}&
-\end{align*}
+\end{flalign*}
 We can make this composition orthogonal with respect to the Blanchfield paring.
 \vspace{0.5cm}\\
 Historical remark:
@@ -893,10 +896,10 @@ Suppose $A$ and $B$ are two symmetric polynomials that are coprime and that $\fo
 \begin{proof}[Idea of proof]
 For any $z$ find an interval $(a_z, b_z)$ such that if $P(z) \in (a_z, b_z)$ and $P(z)A(z) + Q(z)B(z) = 1$, then $Q(z) > 0$, $x(z) = \frac{az + bz}{i}$ is a continues function on $S^1$ approximating $z$ by a polynomial .
 \\??????????????????????????\\
-\begin{align*}
+\begin{flalign*}
-(1, 1) \mapsto \frac{h}{p^k} \mapsto \frac{g\overbar{g}h}{p^k}\\
+(1, 1) \mapsto \frac{h}{p^k} \mapsto \frac{g\overbar{g}h}{p^k}&\\
-g\overbar{g} h + p^k\omega = 1
+g\overbar{g} h + p^k\omega = 1&
-\end{align*}
+\end{flalign*}
 Apply Lemma \ref{L:coprime polynomials} for $A=h$, $B=p^{2k}$. Then, if the assumptions are satisfied, 
 \begin{align*}
 Ph + Qp^{2k} = 1\\
@@ -927,7 +930,7 @@ If $P$ has no roots on $S^1$ then $B(z) > 0$ for all $z$, so the assumptions of
 \begin{theorem}(Matumoto, Conway-Borodzik-Politarczyk)
 Let $K$ be a knot, 
 \begin{align*}
-H_1(\widetilde{X}, \Lambda) \times
+&H_1(\widetilde{X}, \Lambda) \times
 H_1(\widetilde{X}, \Lambda) 
 = \bigoplus_{\substack{k, \xi, \epsilon\\ \xi in S^1}}
 (\quot{\Lambda}{p_{\xi}^k}, \epsilon)^{n_k, \xi, \epsilon} \oplus \bigoplus_{k, \eta}