diff --git a/images/ball_4_pushed_seifert.pdf b/images/ball_4_pushed_seifert.pdf
index f8a5e12..dd1e3e8 100644
Binary files a/images/ball_4_pushed_seifert.pdf and b/images/ball_4_pushed_seifert.pdf differ
diff --git a/images/ball_4_pushed_seifert.pdf_tex b/images/ball_4_pushed_seifert.pdf_tex
index 88308a8..a089784 100644
--- a/images/ball_4_pushed_seifert.pdf_tex
+++ b/images/ball_4_pushed_seifert.pdf_tex
@@ -36,7 +36,7 @@
}%
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\else%
@@ -48,17 +48,16 @@
\global\let\svgwidth\undefined%
\global\let\svgscale\undefined%
\makeatother%
- \begin{picture}(1,0.96340964)%
+ \begin{picture}(1,0.87333181)%
\put(0,0){\includegraphics[width=\unitlength,page=1]{ball_4_pushed_seifert.pdf}}%
- \put(0.1505123,0.68779344){\color[rgb]{0,0,0}\makebox(0,0)[lt]{\begin{minipage}{2.12585306\unitlength}\raggedright $B^4$ \end{minipage}}}%
+ \put(0.33217286,0.77104639){\color[rgb]{0,0,0}\makebox(0,0)[lt]{\begin{minipage}{1.96618474\unitlength}\raggedright $B^4$ \end{minipage}}}%
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+ \put(1.36004492,1.65392014){\color[rgb]{0,0,0}\makebox(0,0)[lt]{\begin{minipage}{0.39774488\unitlength}\raggedright \end{minipage}}}%
\put(0,0){\includegraphics[width=\unitlength,page=2]{ball_4_pushed_seifert.pdf}}%
- \put(1.24884474,1.80672123){\color[rgb]{0,0,0}\makebox(0,0)[lt]{\begin{minipage}{0.43004462\unitlength}\raggedright \end{minipage}}}%
- \put(0.78383734,0.70000247){\color[rgb]{0,0,0}\makebox(0,0)[lt]{\begin{minipage}{0.19056149\unitlength}\raggedright $\Sigma$\end{minipage}}}%
+ \put(0.59916734,0.31598118){\color[rgb]{0,0,0}\makebox(0,0)[lt]{\begin{minipage}{0.90744424\unitlength}\raggedright $g(F) = g_4(K)$ \end{minipage}}}%
+ \put(0.6042564,0.37550965){\color[rgb]{0,0,0}\makebox(0,0)[lt]{\begin{minipage}{0.85985262\unitlength}\raggedright $F \subset B^4$ \end{minipage}}}%
+ \put(0.6549243,0.82850612){\color[rgb]{0,0,0}\makebox(0,0)[lt]{\begin{minipage}{1.03767986\unitlength}\raggedright $S^3$ \end{minipage}}}%
\put(0,0){\includegraphics[width=\unitlength,page=3]{ball_4_pushed_seifert.pdf}}%
- \put(1.34153286,1.59528042){\color[rgb]{0,0,0}\makebox(0,0)[lt]{\begin{minipage}{0.43004462\unitlength}\raggedright \end{minipage}}}%
- \put(0,0){\includegraphics[width=\unitlength,page=4]{ball_4_pushed_seifert.pdf}}%
- \put(0.47141503,0.31803536){\color[rgb]{0,0,0}\makebox(0,0)[lt]{\begin{minipage}{1.74438909\unitlength}\raggedright $g(F) = g_4(K)$ \end{minipage}}}%
- \put(0.49282511,0.21162196){\color[rgb]{0,0,0}\makebox(0,0)[lt]{\begin{minipage}{1.29482674\unitlength}\raggedright $F \subset B^4$ \end{minipage}}}%
- \put(0.82148101,0.91571792){\color[rgb]{0,0,0}\makebox(0,0)[lt]{\begin{minipage}{2.12585306\unitlength}\raggedright $S^3$ \end{minipage}}}%
+ \put(0.81311761,0.76290458){\color[rgb]{0,0,0}\makebox(0,0)[lt]{\begin{minipage}{0.30641817\unitlength}\raggedright $\Sigma$\end{minipage}}}%
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\endgroup%
diff --git a/images/ball_4_pushed_seifert.svg b/images/ball_4_pushed_seifert.svg
index 19eb77c..b40c812 100644
--- a/images/ball_4_pushed_seifert.svg
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@@ -11,9 +11,9 @@
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inkscape:pageshadow="2"
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- transform="matrix(1.2245251,0,0,2.2214098,-1092.0133,979.80704)">$B^4$
- $B^4$ $\Sigma$ $\Sigma$
- $\Sigma$
-
-
-
-
-
+
+
+
+
+
+
+
+
+
+
$g(F) = g_4(K)$ $F \subset B^4$ $S^3$
+ id="flowPara618">$S^3$
+ $\Sigma$
diff --git a/images/seifert_matrix.pdf b/images/seifert_matrix.pdf
index 3b417ab..c11d98a 100644
Binary files a/images/seifert_matrix.pdf and b/images/seifert_matrix.pdf differ
diff --git a/images/seifert_matrix.pdf_tex b/images/seifert_matrix.pdf_tex
index 4851a84..86c0bad 100644
--- a/images/seifert_matrix.pdf_tex
+++ b/images/seifert_matrix.pdf_tex
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\put(0.68037809,0.26129435){\color[rgb]{0,0,0}\makebox(0,0)[lt]{\begin{minipage}{0.12493878\unitlength}\raggedright $\alpha_1$\end{minipage}}}%
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diff --git a/images/seifert_matrix.svg b/images/seifert_matrix.svg
index ee448a3..dddaa5c 100644
--- a/images/seifert_matrix.svg
+++ b/images/seifert_matrix.svg
@@ -16,13 +16,43 @@
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inkscape:export-filename="/Users/Kasia/lectures_on_knot_theory/images/seifert3d.png"
inkscape:export-xdpi="90"
inkscape:export-ydpi="90">
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gradientUnits="userSpaceOnUse"
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$\Sigma$ $\Sigma$
+ y="121.68215"
+ style="font-size:40px;line-height:1.25">
+ y="358.72156"
+ style="font-size:40px;line-height:1.25">
-
-
-
-
-
-
-
+
+
+
+
+
$\alpha_1^+$ $\alpha_1^+$ $\alpha_1$
-
-
-
-
-
-
+ id="flowPara5776"
+ style="font-size:40.00000381px;line-height:1.25">$\alpha_1$
+
+
+
+
$\alpha_1$ $\alpha_1$ $\alpha_2$ $\alpha_2$
+
+
+
diff --git a/lectures_on_knot_theory.tex b/lectures_on_knot_theory.tex
index bcbe6be..bdb5452 100644
--- a/lectures_on_knot_theory.tex
+++ b/lectures_on_knot_theory.tex
@@ -32,6 +32,8 @@
\usepackage{unicode-math}
+
+
\usetikzlibrary{cd}
\hypersetup{
@@ -135,12 +137,12 @@ Let us consider a long exact sequence of cohomology of a pair $(S^3, S^3 \setmin
&\mathbb{Z}
\\
-& H^0(S^3) \ar[u,isomorphic] \to
+& H^0(S^3) \ar[u,isomorphic] \to
&H^0(S^3 \setminus N) \to
\\
\to H^1(S^3, S^3 \setminus N) \to
& H^1(S^3) \to
- & H^1(S^3\setminus N) \to
+ & H^1(S^3\setminus N) \to
\\
& 0 \ar[u,isomorphic]&
\\
@@ -155,13 +157,29 @@ Let us consider a long exact sequence of cohomology of a pair $(S^3, S^3 \setmin
& \mathbb{Z} \ar[u,isomorphic] &\\
\end{tikzcd}
\end{center}
-\[
-H^* (S^3, S^3 \setminus N) \cong H^* (N, \partial N)
-\]
+\begin{align*}
+N \cong & D^2 \times S^1\\
+\partial N \cong & S^1 \times S^1\\
+H^1(N, \partial N) \cong & \mathbb{Z} \oplus \mathbb{Z}
+\end{align*}
+\begin{align*}
+H^* (S^3, S^3 \setminus N) &\cong H^* (N, \partial N)\\
\\
-??????????????
-\\
-
+H^ 1 (S^3\setminus N) &\cong H^1(S^3\setminus K) \cong \mathbb{Z}
+\end{align*}
+\begin{equation*}
+\begin{tikzcd}[row sep=huge]
+H^1(S^3 \setminus K) \arrow[r,] \arrow[d,"\widetilde{\Theta}"] &
+H^1(N \setminus K) \arrow[d,"\Theta"] \\
+{[S^3 \setminus K, S^1]} \arrow[r,]&
+{[N \setminus K, S^1]}
+\end{tikzcd}
+\end{equation*}
+\noindent
+$\Sigma = \widetilde{\Theta}^{-1}(X)$ is a surface, such that $\partial \Sigma = K$, so it is a Seifert surface.
+%
+%
+% Thom isomorphism,
\end{proof}
\begin{definition}
@@ -312,13 +330,39 @@ ${D^2 \overset{g}\longhookrightarrow M}$ such that:
g_{\big| \partial D^2} = f_{\big| \partial D^2.}
\]
\end{lemma}
-\section{}
+\noindent
+Remark: Dehn lemma doesn't hold for dimension four.\\
+Let $M$ be connected, compact three manifold with boundary.
+Suppose $\pi_1(\partial M) \longrightarrow \pi_1(M)$ has non-trivial kernel. Then there exists a map $f: (D^2, \partial D^2) \longrightarrow (M, \partial M)$ such that $f\big| \partial D^2$ is non-trivial loop in $\partial M$
+\begin{theorem}[Sphere theorem]
+Suppose $\pi_1(M) \ne 0$. Then there exists an embedding $f: S^2 \hookrightarrow M$ that is homotopy non-trivial.
+\end{theorem}
+\begin{problem}
+Prove that $S^3 \ K$ is Eilenberg–MacLane space of type $K(\pi, 1)$.
+\end{problem}
+\begin{corollary}
+Suppose $K \subset S^3$ and $\pi_1(S^3 \setminus K)$ is infinite cyclic ($\mathbb{Z})$. Then $K$ is trivial.
+\end{corollary}
+\subsection*{Construction}
+We know that $3$ - sphere can be obtained by gluing two solid tori:
+$S^3 = (D^2 \times S^1) \cup (S^1 \times D^2)$. So the complement of solid torus in $S^3$ is another solid torus.\\
+Take $(z_1, z_2) \in \mathbb{C}$ such that $max(\mid z_1 \mid, \mid z_2, \mid) = 1
+$
+\begin{figure}[h]
+\centering{
+\def\svgwidth{\linewidth}
+\resizebox{0.3\textwidth}{!}{\includegraphics[width=0.3\textwidth]{sphere_as_torus.png}}
+\caption{The complement of solid torus in $S^3$ is another solid torus.}
+\label{fig:sphere_as_tori}
+}
+\end{figure}
+
+
\begin{example}
\begin{align*}
&F: \mathbb{C}^2 \rightarrow \mathbb{C} \text{ a polynomial} \\
&F(0) = 0
\end{align*}
-
\end{example}
????????????
\\
@@ -348,6 +392,7 @@ Figure 8 knot is negative amphichiral.
%
%
%
+
\section{Concordance group \hfill\DTMdate{2019-03-18}}
\begin{definition}
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
@@ -1064,6 +1109,10 @@ H_{p, i} = \quot{\mathbb{Z}[t, t^{-1}]}{p^{k_i} \mathbb{Z} [t, t^{-1}]}
\noindent
The proof is the same as over $\mathbb{Z}$.
\noindent
+%Add NotePrintSaveCiteYour opinionEmailShare
+%Saveliev, Nikolai
+%Lectures on the Topology of 3-Manifolds
+%An Introduction to the Casson Invariant
\end{document}