diff --git a/images/linking_hopf.pdf b/images/linking_hopf.pdf
index 153e57e..9cdd4ec 100644
Binary files a/images/linking_hopf.pdf and b/images/linking_hopf.pdf differ
diff --git a/images/linking_hopf.pdf_tex b/images/linking_hopf.pdf_tex
index 4bb23f7..e828aec 100644
--- a/images/linking_hopf.pdf_tex
+++ b/images/linking_hopf.pdf_tex
@@ -36,7 +36,7 @@
}%
\providecommand\rotatebox[2]{#2}%
\ifx\svgwidth\undefined%
- \setlength{\unitlength}{364.49980969bp}%
+ \setlength{\unitlength}{280.39412611bp}%
\ifx\svgscale\undefined%
\relax%
\else%
@@ -48,12 +48,12 @@
\global\let\svgwidth\undefined%
\global\let\svgscale\undefined%
\makeatother%
- \begin{picture}(1,0.42290942)%
- \put(0.29250579,2.25393507){\color[rgb]{0,0,0}\makebox(0,0)[lt]{\begin{minipage}{0.26372184\unitlength}\raggedright \end{minipage}}}%
- \put(0.29250579,2.25393507){\color[rgb]{0,0,0}\makebox(0,0)[lt]{\begin{minipage}{0.26372184\unitlength}\raggedright \end{minipage}}}%
+ \begin{picture}(1,0.64151204)%
+ \put(0.294701,3.02176327){\color[rgb]{0,0,0}\makebox(0,0)[lt]{\begin{minipage}{0.34282656\unitlength}\raggedright \end{minipage}}}%
+ \put(0.294701,3.02176327){\color[rgb]{0,0,0}\makebox(0,0)[lt]{\begin{minipage}{0.34282656\unitlength}\raggedright \end{minipage}}}%
\put(0,0){\includegraphics[width=\unitlength,page=1]{linking_hopf.pdf}}%
- \put(0.86572181,0.24480623){\color[rgb]{0,0,0}\makebox(0,0)[lt]{\begin{minipage}{0.48025377\unitlength}\raggedright $\Lk(\alpha, \beta) = -1$\\ \end{minipage}}}%
- \put(-0.0000008,0.37342144){\color[rgb]{1,0,0}\makebox(0,0)[lt]{\begin{minipage}{0.21983917\unitlength}\raggedright $\alpha$\\ \end{minipage}}}%
- \put(0.73747224,0.40373095){\color[rgb]{0,0,1}\makebox(0,0)[lt]{\begin{minipage}{0.21983917\unitlength}\raggedright $\beta$\\ \end{minipage}}}%
+ \put(0.40133526,0.02351275){\color[rgb]{0,0,0}\makebox(0,0)[lt]{\begin{minipage}{0.62430836\unitlength}\raggedright $\Lk(\alpha, \beta) = -1$\\ \end{minipage}}}%
+ \put(-0.00000051,0.59179048){\color[rgb]{1,0,0}\makebox(0,0)[lt]{\begin{minipage}{0.14230468\unitlength}\raggedright $\alpha$\\ \end{minipage}}}%
+ \put(0.8318937,0.49629103){\color[rgb]{0,0,1}\makebox(0,0)[lt]{\begin{minipage}{0.16825112\unitlength}\raggedright $\beta$\\ \end{minipage}}}%
\end{picture}%
\endgroup%
diff --git a/images/linking_hopf.svg b/images/linking_hopf.svg
index 23e527b..b1fff29 100644
--- a/images/linking_hopf.svg
+++ b/images/linking_hopf.svg
@@ -10,9 +10,9 @@
xmlns="http://www.w3.org/2000/svg"
xmlns:sodipodi="http://sodipodi.sourceforge.net/DTD/sodipodi-0.dtd"
xmlns:inkscape="http://www.inkscape.org/namespaces/inkscape"
- width="128.58743mm"
- height="54.380836mm"
- viewBox="0 0 455.62479 192.68801"
+ width="98.916817mm"
+ height="63.456329mm"
+ viewBox="0 0 350.49268 224.84528"
id="svg2"
version="1.1"
inkscape:version="0.92.2 5c3e80d, 2017-08-06"
@@ -213,10 +213,10 @@
is_visible="true" />
+ originx="-431.95973"
+ originy="115.31789" />
@@ -1845,7 +1845,7 @@
inkscape:label="Layer 1"
inkscape:groupmode="layer"
id="layer1"
- transform="translate(-401.97736,-942.83463)">
+ transform="translate(-431.9597,-942.83463)">
$\Lk(\alpha, \beta) = -1$
+ originx="-381.21121"
+ originy="115.16802" />
@@ -1877,7 +1877,7 @@
inkscape:label="Layer 1"
inkscape:groupmode="layer"
id="layer1"
- transform="translate(-390.97626,-940.39913)">
+ transform="translate(-381.21127,-940.39913)">
\shortstack{$\Lk(\alpha, \beta) = 3$}
+
+
+
diff --git a/images/unknot.png b/images/unknot.png
index db7008e..2f7ea3e 100644
Binary files a/images/unknot.png and b/images/unknot.png differ
diff --git a/images/unknot.svg b/images/unknot.svg
new file mode 100644
index 0000000..502ab7d
--- /dev/null
+++ b/images/unknot.svg
@@ -0,0 +1,119 @@
+
+
+
+
diff --git a/lec_04_03.tex b/lec_04_03.tex
index eeb7dda..d2e89bf 100644
--- a/lec_04_03.tex
+++ b/lec_04_03.tex
@@ -2,7 +2,7 @@
%\begin{theorem}
%For any knot $K \subset S^3$ there exists a connected, compact and orientable surface $\Sigma(K)$ such that $\partial \Sigma(K) = K$
%\end{theorem}
-\begin{proof}(Theorem \ref{theo:Seifert})\\
+\begin{proof}(\Cref{theo:Seifert})\\
Let $K \in S^3$ be a knot and $N = \nu(K)$ be its tubular neighbourhood. Because $K$ and $N$ are homotopy equivalent, we get:
\begin{align*}
H^1(S^3 \setminus N ) \cong H^1(S^3 \setminus K).
diff --git a/lec_25_02.tex b/lec_25_02.tex
index 3e1c85b..5c7a482 100644
--- a/lec_25_02.tex
+++ b/lec_25_02.tex
@@ -1,69 +1,104 @@
\begin{definition}
-A knot $K$ in $S^3$ is a smooth (PL - smooth) embedding of a circle $S^1$ in $S^3$:
-\begin{align*}
-\varphi: S^1 \hookrightarrow S^3
-\end{align*}
+ A knot $K$ in $S^3$ is a smooth (PL - smooth)
+ embedding of a circle $S^1$ in $S^3$:
+ \[
+ \varphi: S^1 \hookrightarrow S^3
+ \]
\end{definition}
+
\noindent
-Usually we think about a knot as an image of an embedding: $K = \varphi(S^1)$.
-Some basic examples and counterexamples are shown respectively in \autoref{fig:unknot} and \autoref{fig:notknot}.
-\begin{example}
+Usually we think about a knot
+as an image of an embedding:
+$K = \varphi(S^1)$.
+Some basic examples and counterexamples
+are shown respectively in
+\autoref{fig:unknot} and
+\autoref{fig:notknot}.
+
\begin{figure}[h]
-\includegraphics[width=0.08\textwidth]
-{unknot.png}
-\caption{Knots examples: unknot (left) and trefoil (right).}
-\label{fig:unknot}
+ \centering
+ \begin{subfigure}{0.3\textwidth}
+ \centering
+ \includegraphics[width=0.5\textwidth]
+ {unknot.png}
+ \end{subfigure}
+ \begin{subfigure}{0.3\textwidth}
+ \centering
+ \includegraphics[width=0.5\textwidth]
+ {trefoil.png}
+ \end{subfigure}
+ \caption{Knots examples:
+ unknot (left) and trefoil (right).}
+ \label{fig:unknot}
\end{figure}
\begin{figure}[h]
-\includegraphics[width=0.08\textwidth]
-{unknot.png}
-\caption{Knots examples: unknot (left) and trefoil (right).}
-\label{fig:notknot}
+ \centering
+ \begin{subfigure}{0.3\textwidth}
+ \centering
+ \includegraphics[width=0.5\textwidth]
+ {not_injective_knot.png}
+ \end{subfigure}
+ \begin{subfigure}{0.3\textwidth}
+ \centering
+ \includegraphics[width=0.5\textwidth]
+ {not_smooth_knot.png}
+ \end{subfigure}
+ \caption{Not-knots examples:
+ an image of
+ a function ${S^1\longrightarrow S^3}$
+ that isn't injective (left) and
+ of a function
+ that isn't smooth (right).}
+ \label{fig:notknot}
\end{figure}
-
-
-\begin{itemize}
-\item
-Knots:
-\includegraphics[width=0.08\textwidth]{unknot.png} (unknot),
-\includegraphics[width=0.08\textwidth]{trefoil.png} (trefoil).
-\item
-Not knots:
-\includegraphics[width=0.12\textwidth]{not_injective_knot.png}
-(it is not an injection),
-\includegraphics[width=0.08\textwidth]{not_smooth_knot.png}
-(it is not smooth).
-\end{itemize}
-\end{example}
\begin{definition}
-%\hfill\\
-Two knots $K_0 = \varphi_0(S^1)$, $K_1 = \varphi_1(S^1)$ are equivalent if the embeddings $\varphi_0$ and $\varphi_1$ are isotopic, that is there exists a continues function
-\begin{align*}
-&\Phi: S^1 \times [0, 1] \hookrightarrow S^3, \\
-&\Phi(x, t) = \Phi_t(x)
-\end{align*}
- such that $\Phi_t$ is an embedding for any $t \in [0,1]$, $\Phi_0 = \varphi_0$ and
-$\Phi_1 = \varphi_1$.
+ Two knots $K_0 = \varphi_0(S^1)$,
+ $K_1 = \varphi_1(S^1)$
+ are equivalent if the embeddings
+ $\varphi_0$ and $\varphi_1$ are isotopic,
+ that is there exists a continues function
+ \begin{align*}
+ &\Phi: S^1 \times
+ [0, 1] \hookrightarrow S^3, \\
+ &\Phi(x, t) = \Phi_t(x)
+ \end{align*}
+ such that
+ $\Phi_t$ is an embedding
+ for any $t \in [0,1]$,
+ $\Phi_0 = \varphi_0$ and
+ $\Phi_1 = \varphi_1$.
\end{definition}
\begin{theorem}
-Two knots $K_0$ and $K_1$ are isotopic if and only if they are ambient isotopic, i.e. there exists a family of self-diffeomorphisms $\Psi = \{\psi_t: t \in [0, 1]\}$ such that:
-\begin{align*}
-&\psi(t) = \psi_t \text{ is continius on $t\in [0,1]$},\\
-&\psi_t: S^3 \hookrightarrow S^3,\\
-& \psi_0 = id ,\\
-& \psi_1(K_0) = K_1.
-\end{align*}
+ Two knots $K_0$ and $K_1$ are isotopic
+ if and only if they are ambient isotopic,
+ i.e. there exists a family of self-diffeomorphisms
+ $\Psi = \{\psi_t: t \in [0, 1]\}$ such that:
+ \begin{align*}
+ &\psi(t) = \psi_t
+ \text{ is continius on
+ $t\in [0,1]$},\\
+ &\psi_t: S^3 \hookrightarrow S^3,\\
+ & \psi_0 = id ,\\
+ & \psi_1(K_0) = K_1.
+ \end{align*}
\end{theorem}
\begin{definition}
-A knot is trivial (unknot) if it is equivalent to an embedding $\varphi(t) = (\cos t, \sin t, 0)$, where $t \in [0, 2 \pi] $ is a parametrisation of $S^1$.
+ A knot is trivial (unknot) if it is equivalent
+ to an embedding
+ $\varphi(t) = (\cos t, \sin t, 0)$,
+ where $t \in [0, 2 \pi] $
+ is a parametrisation of $S^1$.
\end{definition}
\begin{definition}
-A link with k - components is a (smooth) embedding of $\overbrace{S^1 \sqcup \ldots \sqcup S^1}^k$ in $S^3$.
+ A link with $k$ - components is a
+ (smooth) embedding of
+ $\overbrace{S^1 \sqcup \ldots \sqcup S^1}^k$
+ in $S^3$.
\end{definition}
\begin{example}
@@ -85,20 +120,31 @@ a Borromean link:
%
%
%
-\begin{definition}
-A link diagram $D_{\pi}$ is a picture over projection $\pi$ of a link $L$ in $\mathbb{R}^3$($S^3$) to $\mathbb{R}^2$ ($S^2$) such that:
-\begin{enumerate}[label={(\arabic*)}]
-\item
-$D_{\pi |_L}$ is non degenerate: \includegraphics[width=0.05\textwidth]{LinkDiagram1.png},
-\item the double points are not degenerate: \includegraphics[width=0.03\textwidth]{LinkDiagram2.png},
-\item there are no triple point: \includegraphics[width=0.05\textwidth]{LinkDiagram3.png}.
-\end{enumerate}
+\begin{definition}\label{def:link_diagram}
+ A link diagram $D_{\pi}$ is a picture
+ over projection $\pi$ of a link $L$ in
+ $\mathbb{R}^3$($S^3$) to
+ $\mathbb{R}^2$ ($S^2$) such that:
+ \begin{enumerate}[label={(\arabic*)}]
+ \item
+ $D_{\pi |_L}$ is non degenerate,
+ \item
+ the double points are not degenerate,
+ \item there are no triple point.
+ \end{enumerate}
\end{definition}
+\noindent
+By \Cref{def:link_diagram} the following pictures can not be a part of a diagram:
+\includegraphics[width=0.05\textwidth]{LinkDiagram1.png},
+\includegraphics[width=0.03\textwidth]{LinkDiagram2.png},
+\includegraphics[width=0.05\textwidth]{LinkDiagram3.png}.
+
+
\noindent
There are under- and overcrossings (tunnels and bridges) on a link diagrams with an obvious meaning.
-\begin{fact}
+\begin{lemma}
Every link admits a link diagram.
-\end{fact}
+\end{lemma}
\noindent
Let $D$ be a diagram of an oriented link (to each component of a link we add an arrow in the diagram).
@@ -108,15 +154,17 @@ $\left(\PICorientpluscross\right)$, called a positive crossing, and left-handed
\subsection{Reidemeister moves}
A Reidemeister move is one of the three types of operation on a link diagram as shown below:
\begin{enumerate}[label=\Roman*]
-\item\hfill\\
-\includegraphics[width=0.6\textwidth]{rm1.png},
-\item\hfill\\\includegraphics[width=0.6\textwidth]{rm2.png},
-\item\hfill\\\includegraphics[width=0.4\textwidth]{rm3.png}.
+ \item\hfill\\
+ \includegraphics[width=0.6\textwidth]{rm1.png},
+ \item\hfill\\\includegraphics[width=0.6\textwidth]{rm2.png},
+ \item\hfill\\\includegraphics[width=0.4\textwidth]{rm3.png}.
\end{enumerate}
\begin{theorem} [Reidemeister, 1927 ]
-Two diagrams of the same link can be
-deformed into each other by a finite sequence of Reidemeister moves (and isotopy of the plane).
+ Two diagrams of the same link can
+ be deformed into each other by a finite
+ sequence of Reidemeister moves
+ (and isotopy of the plane).
\end{theorem}
%
%
@@ -134,169 +182,244 @@ deformed into each other by a finite sequence of Reidemeister moves (and isotopy
\noindent
Let $D$ be an oriented diagram of a link $L$. We change the diagram by smoothing each crossing:
\begin{align*}
-\PICorientpluscross \mapsto \PICorientLRsplit,\\
-\PICorientminuscross \mapsto \PICorientLRsplit.
+ \PICorientpluscross \mapsto
+ \PICorientLRsplit,\\
+ \PICorientminuscross \mapsto
+ \PICorientLRsplit.
\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 $\partial \Sigma = L$.\\
+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]
-\fontsize{15}{10}\selectfont
-\centering{
-\def\svgwidth{\linewidth}
-\resizebox{0.8\textwidth}{!}{\input{images/seifert_alg.pdf_tex}}
-\caption{Constructing a Seifert surface.}
-\label{fig:SeifertAlg}
-}
+ \fontsize{15}{10}\selectfont
+ \centering{
+ \def\svgwidth{\linewidth}
+ \resizebox{0.8\textwidth}{!}
+ {\input{images/seifert_alg.pdf_tex}}
+ \caption{Constructing a Seifert surface.}
+ \label{fig:SeifertAlg}
+ }
\end{figure}
\noindent
Note: the obtained surface isn't unique and in general 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 $D_2$. Then we glue both components on the boundaries: $\partial D_1$ and $\partial D_2$.
\begin{figure}[h]
-\begin{center}
-\includegraphics[width=0.6\textwidth]{seifert_connect.png}
-\end{center}
-\caption{Connecting two surfaces.}
-\label{fig:SeifertConnect}
+ \centering
+ \includegraphics[width=0.6\textwidth]
+ {seifert_connect.png}
+ \caption{Connecting two surfaces.}
+ \label{fig:SeifertConnect}
\end{figure}
-\begin{theorem}[Seifert]
-\label{theo:Seifert}
-Every link in $S^3$ bounds a surface $\Sigma$ that is compact, connected and orientable. Such a surface is called a Seifert surface.
+\begin{theorem}[Seifert]\label{theo:Seifert}
+ 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{12}{10}\selectfont
-\centering{
-\def\svgwidth{\linewidth}
-\resizebox{1\textwidth}{!}{\input{images/torus_1_2_3.pdf_tex}}
-\caption{Genus of an orientable surface.}
-\label{fig:genera}
-}
+ \fontsize{12}{10}\selectfont
+ \centering
+ \def\svgwidth{\linewidth}
+ \resizebox{1\textwidth}{!}{
+ \input{images/torus_1_2_3.pdf_tex}}
+ \caption{Genus of an orientable surface.}
+ \label{fig:genera}
\end{figure}
%
%
\begin{definition}
-The three genus $g_3(K)$ ($g(K)$) of a knot $K$ is the minimal genus of a Seifert surface $\Sigma$ for $K$.
+ The three genus $g_3(K)$ ($g(K)$)
+ of a knot $K$ is the minimal genus
+ of a Seifert surface $\Sigma$ for $K$.
\end{definition}
\begin{corollary}
-A knot $K$ is trivial if and only $g_3(K) = 0$.
+ A knot $K$ is trivial if and only
+ $g_3(K) = 0$.
\end{corollary}
\noindent
-Remark: there are knots that admit non isotopic Seifert surfaces of minimal genus (András Juhász, 2008).
+Remark: there are knots that admit non isotopic
+Seifert surfaces of minimal genus
+(András Juhász, 2008).
\begin{definition}
-Suppose $\alpha$ and $\beta$ are two simple closed curves in $\mathbb{R}^3$.
-On a diagram $L$ consider all crossings between $\alpha$ and $\beta$. Let $N_+$ be the number of positive crossings, $N_-$ - negative. Then the linking number: $\Lk(\alpha, \beta) = \frac{1}{2}(N_+ - N_-)$.
-\end{definition}
-\begin{definition}
-\label{def:lk_via_homo}
-Let $\alpha$ and $\beta$ be two disjoint simple cross curves in $S^3$.
-Let $\nu(\beta)$ be a tubular neighbourhood of $\beta$. The linking number can be interpreted via first homology group, where $\Lk(\alpha, \beta)$ is equal to evaluation of $\alpha$ as element of first homology group of the complement of $\beta$:
-\[
-\alpha \in H_1(S^3 \setminus \nu(\beta), \mathbb{Z}) \cong \mathbb{Z}.\]
+ Suppose $\alpha$ and $\beta$ are two
+ simple closed curves in $\mathbb{R}^3$.
+ On a diagram $L$ consider all crossings
+ between $\alpha$ and $\beta$.
+ Let $N_+$ be the number
+ of positive crossings,
+ $N_-$ - negative.
+ Then the linking number:
+ $\Lk(\alpha, \beta) =
+ \frac{1}{2}(N_+ - N_-)$.
+\end{definition}
+
+\begin{definition}\label{def:lk_via_homo}
+ Let $\alpha$ and $\beta$ be
+ two disjoint simple closed curves in $S^3$.
+ Let $\nu(\beta)$ be a tubular
+ neighbourhood of $\beta$.
+ The linking number can be interpreted
+ via first homology group, where
+ $\Lk(\alpha, \beta)$ is equal
+ to evaluation of $\alpha$ as element
+ of first homology group
+ of the complement of $\beta$:
+ \[
+ \alpha \in H_1(S^3 \setminus
+ \nu(\beta), \mathbb{Z})
+ \cong \mathbb{Z}.
+ \]
\end{definition}
-\begin{example}
-\begin{itemize}
-\item
-A Hopf link:
\begin{figure}[h]
-\fontsize{20}{10}\selectfont
-\centering{
-\def\svgwidth{\linewidth}
-\resizebox{0.4\textwidth}{!}{\input{images/linking_hopf.pdf_tex}},
-}
+ \fontsize{10}{8}\selectfont
+ \centering
+ \def\svgwidth{\linewidth}
+ \resizebox{\textwidth}{!}{
+% \centering
+ \begin{subfigure}{0.3\textwidth}
+ \centering
+ \def\svgwidth{\linewidth}
+ \resizebox{1\textwidth}{!}{
+ \input{images/linking_torus_6_2.pdf_tex}
+ }
+ \end{subfigure}
+ \begin{subfigure}{0.3\textwidth}
+ \centering
+ \def\svgwidth{\linewidth}
+ \resizebox{1\textwidth}{!}{
+ \input{images/linking_hopf.pdf_tex}
+ }
+ \end{subfigure}
+ }
+ \vspace*{10mm}
+ \caption{
+ Linking number of a Hopf link (left)
+ and a torus link $T(6, 2)$ (right).
+ }
+ \label{fig:unknot}
\end{figure}
-\item
-$T(6, 2)$ link:
-\begin{figure}[h]
-\fontsize{20}{10}\selectfont
-\centering{
-\def\svgwidth{\linewidth}
-\resizebox{0.4\textwidth}{!}{\input{images/linking_torus_6_2.pdf_tex}}.
-}
-\end{figure}
-\end{itemize}
-\end{example}
+
\begin{fact}
$
g_3(\Sigma) = \frac{1}{2} b_1 (\Sigma) =
\frac{1}{2} \dim_{\mathbb{R}}H_1(\Sigma, \mathbb{R}),
$
-where $b_1$ is first Betti number of $\Sigma$.
+where $b_1$ is first Betti number of a surface $\Sigma$.
\end{fact}
\subsection{Seifert matrix}
-Let $L$ be a link and $\Sigma$ be an oriented Seifert surface for $L$. Choose a basis for $H_1(\Sigma, \mathbb{Z})$ consisting of simple closed curves $\alpha_1, \dots, \alpha_n$.
-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 $\alpha_i^+$ 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.
+Let $L$ be a link and
+$\Sigma$ be an oriented
+Seifert surface for $L$.
+Choose a basis for
+$H_1(\Sigma, \mathbb{Z})$
+consisting of simple closed curves
+$\alpha_1, \dots, \alpha_n$.
+
+\noindent
+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
+$\alpha_i^+$ don't intersect the surface.
+
+\noindent
+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]
-\fontsize{20}{10}\selectfont
-\centering{
-\def\svgwidth{\linewidth}
-\resizebox{0.8\textwidth}{!}{\input{images/seifert_matrix.pdf_tex}}
-}
+ \fontsize{20}{10}\selectfont
+ \centering
+ \def\svgwidth{\linewidth}
+ \resizebox{0.8\textwidth}{!}{
+ \input{images/seifert_matrix.pdf_tex}
+ }
+ \caption{
+ A basis $\alpha_1, \alpha_2$
+ of the first homology
+ group of a Seifert surface
+ and a copy of
+ element $\alpha_1$ pushed up
+ along vector normal to the Seifert surface.
+ }
+ \label{fig:alpha_plus}
\end{figure}
\begin{theorem}
-The Seifert matrices $S_1$ and $S_2$ for the same link $L$ are S-equivalent, that is, $S_2$ can be obtained from $S_1$ by a sequence of following moves:
-\begin{enumerate}[label={(\arabic*)}]
-
-\item
-$V \rightarrow AVA^T$, where $A$ is a matrix with integer coefficients,
-
-\item
-
-$V \rightarrow
-\begin{pmatrix}
- \begin{array}{c|c}
- V &
- \begin{matrix}
- \ast & 0 \\
- \sdots & \sdots\\
- \ast & 0
- \end{matrix} \\
- \hline
- \begin{matrix}
- \ast & \dots & \ast\\
- 0 & \dots & 0
- \end{matrix}
- &
- \begin{matrix}
- 0 & 0\\
- 1 & 0
- \end{matrix}
- \end{array}
-\end{pmatrix} \quad$
-or
-$\quad
-V \rightarrow
-\begin{pmatrix}
- \begin{array}{c|c}
- V &
- \begin{matrix}
- \ast & 0 \\
- \sdots & \sdots\\
- \ast & 0
- \end{matrix} \\
- \hline
- \begin{matrix}
- \ast & \dots & \ast\\
- 0 & \dots & 0
- \end{matrix}
- &
- \begin{matrix}
- 0 & 1\\
- 0 & 0
- \end{matrix}
- \end{array}
-\end{pmatrix},$
-\item
-inverse of (2).
-
-\end{enumerate}
+ The Seifert matrices $S_1$ and $S_2$
+ for the same link $L$ are S-equivalent,
+ that is, $S_2$ can be obtained from
+ $S_1$ by a sequence of following moves:
+ \begin{enumerate}[label={(\arabic*)}]
+ \item
+ $V \rightarrow AVA^T$,
+ where $A$ is a matrix
+ with integer coefficients,
+ \item
+ $V \rightarrow
+ \begin{pmatrix}
+ \begin{array}{c|c}
+ V &
+ \begin{matrix}
+ \ast & 0 \\
+ \sdots & \sdots\\
+ \ast & 0
+ \end{matrix} \\
+ \hline
+ \begin{matrix}
+ \ast & \dots & \ast\\
+ 0 & \dots & 0
+ \end{matrix}
+ &
+ \begin{matrix}
+ 0 & 0\\
+ 1 & 0
+ \end{matrix}
+ \end{array}
+ \end{pmatrix} \quad$
+ or
+ $\quad
+ V \rightarrow
+ \begin{pmatrix}
+ \begin{array}{c|c}
+ V &
+ \begin{matrix}
+ \ast & 0 \\
+ \sdots & \sdots\\
+ \ast & 0
+ \end{matrix} \\
+ \hline
+ \begin{matrix}
+ \ast & \dots & \ast\\
+ 0 & \dots & 0
+ \end{matrix}
+ &
+ \begin{matrix}
+ 0 & 1\\
+ 0 & 0
+ \end{matrix}
+ \end{array}
+ \end{pmatrix},$
+ \item
+ inverse of (2).
+ \end{enumerate}
\end{theorem}
diff --git a/lec_mess.tex b/lec_mess.tex
new file mode 100644
index 0000000..9541a40
--- /dev/null
+++ b/lec_mess.tex
@@ -0,0 +1,48 @@
+
+
+
+
+\begin{fact}[Milnor Singular Points of Complex Hypersurfaces]
+\end{fact}
+%\end{comment}
+\noindent
+An oriented knot is called negative amphichiral if the mirror image $m(K)$ of $K$ is equivalent the reverse knot of $K$: $K^r$. \\
+\begin{problem}
+Prove that if $K$ is negative amphichiral, then $K \# K = 0$ in
+$\mathscr{C}$.
+%
+%\\
+%Hint: $ -K = m(K)^r = (K^r)^r = K$
+\end{problem}
+\begin{example}
+Figure 8 knot is negative amphichiral.
+\end{example}
+%
+%
+\begin{theorem}
+Let $H_p$ be a $p$ - torsion part of $H$. There exists an orthogonal decomposition of $H_p$:
+\[
+H_p = H_{p, 1} \oplus \dots \oplus H_{p, r_p}.
+\]
+$H_{p, i}$ is a cyclic module:
+\[
+H_{p, i} = \quot{\mathbb{Z}[t, t^{-1}]}{p^{k_i} \mathbb{Z} [t, t^{-1}]}
+\]
+\end{theorem}
+\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
+
+\begin{figure}[h]
+\fontsize{10}{10}\selectfont
+\centering{
+\def\svgwidth{\linewidth}
+\resizebox{0.5\textwidth}{!}{\input{images/ball_4_alpha_beta.pdf_tex}}
+}
+%\caption{Sketch for Fact %%\label{fig:concordance_m}
+\end{figure}
diff --git a/lectures_on_knot_theory.pdf b/lectures_on_knot_theory.pdf
index 040fc66..001ffc2 100644
Binary files a/lectures_on_knot_theory.pdf and b/lectures_on_knot_theory.pdf differ
diff --git a/lectures_on_knot_theory.tex b/lectures_on_knot_theory.tex
index 097d940..8728ca3 100644
--- a/lectures_on_knot_theory.tex
+++ b/lectures_on_knot_theory.tex
@@ -9,7 +9,9 @@
\usepackage[english]{babel}
-\usepackage{caption}
+\usepackage[margin=1 cm]{caption}
+\usepackage{subcaption}
+%\usepackage{cleveref} - after hyperref
\usepackage{comment}
\usepackage{csquotes}
@@ -21,6 +23,7 @@
\usepackage{graphicx}
\usepackage{hyperref}
+\usepackage[nameinlink]{cleveref}
\usepackage{mathtools}
@@ -28,6 +31,7 @@
\usepackage[section]{placeins}
\usepackage[pdf]{pstricks}
+%\usepackage{subcaption} % added after caption
\usepackage{tikz}
\usepackage{titlesec}
@@ -127,7 +131,7 @@
\begin{document}
\tableofcontents
-%\newpage
+\newpage
%\input{myNotes}
\section{Basic definitions
@@ -214,7 +218,7 @@ Surgery \hfill\DTMdate{2019-06-03}}
\texorpdfstring{
\hfill\DTMdate{2019-06-17}}
{}}
-\input{mess.tex}
+\input{lec_mess.tex}
\end{document}