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new file mode 100644
index 0000000..1ec72f7
--- /dev/null
+++ b/images/BorromeanRings.svg
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diff --git a/images/all.svg b/images/all.svg
new file mode 100644
index 0000000..3c7e4d6
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diff --git a/images/ball_4.pdf b/images/ball_4.pdf
index 1fc18c3..b624733 100644
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diff --git a/images/ball_4.pdf_tex b/images/ball_4.pdf_tex
index 120468c..88b0b2f 100644
--- a/images/ball_4.pdf_tex
+++ b/images/ball_4.pdf_tex
@@ -36,7 +36,7 @@
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diff --git a/images/ink_diag.pdf_tex b/images/ink_diag.pdf_tex
deleted file mode 100644
index b540640..0000000
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diff --git a/images/link_diagram.pdf_tex b/images/link_diagram.pdf_tex
deleted file mode 100644
index 9e90e69..0000000
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index 10fd06c..f35eb1d 100644
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inkscape:version="0.91 r13725"
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inkscape:export-filename="/Users/Kasia/lectures_on_knot_theory/images/seifert3d.png"
inkscape:export-xdpi="90"
inkscape:export-ydpi="90">
@@ -213,10 +213,10 @@
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index 63b89eb..0f730cf 100644
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- \put(2.99321728,-2.29952283){\color[rgb]{0,0,0}\makebox(0,0)[lt]{\begin{minipage}{0.23561194\unitlength}\raggedright \end{minipage}}}%
- \put(3.1713988,-2.26859856){\color[rgb]{0,0,0}\makebox(0,0)[lt]{\begin{minipage}{0.02797896\unitlength}\raggedright \end{minipage}}}%
- \put(3.11691337,-2.27448885){\color[rgb]{0,0,0}\makebox(0,0)[lt]{\begin{minipage}{0.23855692\unitlength}\raggedright \end{minipage}}}%
- \put(0.33374803,0.74723349){\color[rgb]{0,0,0}\makebox(0,0)[lb]{\smash{ }}}%
- \put(0,0){\includegraphics[width=\unitlength,page=1]{torus_1_2_3_v2.pdf}}%
- \put(0.2866221,0.2343449){\color[rgb]{0,0,0}\makebox(0,0)[lt]{\begin{minipage}{0.16212198\unitlength}\raggedright genus $0$\\ \end{minipage}}}%
- \put(0.8744435,0.23989663){\color[rgb]{0,0,0}\makebox(0,0)[lt]{\begin{minipage}{0.1438923\unitlength}\raggedright genus $2$\\ \end{minipage}}}%
- \put(0.2866221,0.08627117){\color[rgb]{0,0,0}\makebox(0,0)[lt]{\begin{minipage}{0.16212198\unitlength}\raggedright genus $1$\\ \end{minipage}}}%
- \put(0.8744435,0.06347455){\color[rgb]{0,0,0}\makebox(0,0)[lt]{\begin{minipage}{0.1438923\unitlength}\raggedright genus $3$\\ \end{minipage}}}%
- \put(0,0){\includegraphics[width=\unitlength,page=2]{torus_1_2_3_v2.pdf}}%
- \end{picture}%
-\endgroup%
diff --git a/lectures_on_knot_theory.tex b/lectures_on_knot_theory.tex
index 773992e..385117f 100644
--- a/lectures_on_knot_theory.tex
+++ b/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]
-\fontsize{15}{10}\selectfont
+\begin{figure}[h]
+\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*}
-&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
-A = V + V^T\\
-&H_1(Y, \mathbb{Z}) \cong \quot{\mathbb{Z}^n}{A\mathbb{Z}}\\
-&A \longrightarrow BAC^T \quad \text{Smith normal form}
-\end{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
+A = V + V^T&
+\\
+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*}
-&\xi \in S^1 \setminus \{ \pm 1\}
+\begin{flalign*}
+\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)(1 - \xi^{-1}) t^{-1}&\\
+\xi \in \mathbb{R} \setminus \{ \pm 1\}
\quad
-q_{\xi} = (t - \xi)(1 - \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)(1 - \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}}
( \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{align*}
+(\quot{\Lambda}{q_{\xi}^l})^{n_l, \xi}&
+\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*}
-(1, 1) \mapsto \frac{h}{p^k} \mapsto \frac{g\overbar{g}h}{p^k}\\
-g\overbar{g} h + p^k\omega = 1
-\end{align*}
+\begin{flalign*}
+(1, 1) \mapsto \frac{h}{p^k} \mapsto \frac{g\overbar{g}h}{p^k}&\\
+g\overbar{g} h + p^k\omega = 1&
+\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}