diff --git a/images/dehn_twist.pdf b/images/dehn_twist.pdf
index 44a5e65..f518c60 100644
Binary files a/images/dehn_twist.pdf and b/images/dehn_twist.pdf differ
diff --git a/images/dehn_twist.svg b/images/dehn_twist.svg
index 047c40e..d0c3aa0 100644
--- a/images/dehn_twist.svg
+++ b/images/dehn_twist.svg
@@ -20,6 +20,18 @@
sodipodi:docname="dehn_twist.svg">
+
+
+
+
+
+ d="m 111.74936,64.635353 c -1.50857,8.628051 -9.97194,8.071102 -15.333563,10.057856 -7.832514,2.902346 -6.903247,10.409815 -7.997307,8.795788 -3.588928,2.753658 2.386444,-14.418694 -8.570427,-12.985964 -5.319088,0.695529 -4.040443,4.850942 -8.395047,4.559546 -4.743437,-0.305985 -7.689258,-7.475306 -7.689258,-10.427226 0,-2.95192 2.685486,-5.624383 7.027329,-7.558865 4.341844,-1.934483 10.340044,-3.130985 16.965474,-3.130985 6.625429,0 12.623629,1.196502 16.965469,3.130985 4.34184,1.934482 7.02733,4.606945 7.02733,7.558865 z"
+ style="opacity:1;fill:none;fill-opacity:1;fill-rule:evenodd;stroke:url(#radialGradient1721);stroke-width:1.06069636;stroke-linejoin:bevel;stroke-miterlimit:4;stroke-dasharray:none;stroke-dashoffset:0;stroke-opacity:0.97073173" />
image/svg+xml
-
+
diff --git a/images/torus_mu_lambda.pdf b/images/torus_mu_lambda.pdf
new file mode 100644
index 0000000..25675b1
Binary files /dev/null and b/images/torus_mu_lambda.pdf differ
diff --git a/images/torus_mu_lambda.pdf_tex b/images/torus_mu_lambda.pdf_tex
new file mode 100644
index 0000000..0b01573
--- /dev/null
+++ b/images/torus_mu_lambda.pdf_tex
@@ -0,0 +1,61 @@
+%% Creator: Inkscape inkscape 0.92.2, www.inkscape.org
+%% PDF/EPS/PS + LaTeX output extension by Johan Engelen, 2010
+%% Accompanies image file 'torus_mu_lambda.pdf' (pdf, eps, ps)
+%%
+%% To include the image in your LaTeX document, write
+%% \input{.pdf_tex}
+%% instead of
+%% \includegraphics{.pdf}
+%% To scale the image, write
+%% \def\svgwidth{}
+%% \input{.pdf_tex}
+%% instead of
+%% \includegraphics[width=]{.pdf}
+%%
+%% Images with a different path to the parent latex file can
+%% be accessed with the `import' package (which may need to be
+%% installed) using
+%% \usepackage{import}
+%% in the preamble, and then including the image with
+%% \import{}{.pdf_tex}
+%% Alternatively, one can specify
+%% \graphicspath{{/}}
+%%
+%% For more information, please see info/svg-inkscape on CTAN:
+%% http://tug.ctan.org/tex-archive/info/svg-inkscape
+%%
+\begingroup%
+ \makeatletter%
+ \providecommand\color[2][]{%
+ \errmessage{(Inkscape) Color is used for the text in Inkscape, but the package 'color.sty' is not loaded}%
+ \renewcommand\color[2][]{}%
+ }%
+ \providecommand\transparent[1]{%
+ \errmessage{(Inkscape) Transparency is used (non-zero) for the text in Inkscape, but the package 'transparent.sty' is not loaded}%
+ \renewcommand\transparent[1]{}%
+ }%
+ \providecommand\rotatebox[2]{#2}%
+ \ifx\svgwidth\undefined%
+ \setlength{\unitlength}{185.50670233bp}%
+ \ifx\svgscale\undefined%
+ \relax%
+ \else%
+ \setlength{\unitlength}{\unitlength * \real{\svgscale}}%
+ \fi%
+ \else%
+ \setlength{\unitlength}{\svgwidth}%
+ \fi%
+ \global\let\svgwidth\undefined%
+ \global\let\svgscale\undefined%
+ \makeatother%
+ \begin{picture}(1,0.62539361)%
+ \put(1.05585685,2.21258294){\color[rgb]{0,0,0}\makebox(0,0)[lt]{\begin{minipage}{0.51818377\unitlength}\raggedright \end{minipage}}}%
+ \put(0,0){\includegraphics[width=\unitlength,page=1]{torus_mu_lambda.pdf}}%
+ \put(0.89358894,0.61469018){\color[rgb]{0,0,0}\makebox(0,0)[lt]{\begin{minipage}{0.10667827\unitlength}\raggedright $\lambda$\end{minipage}}}%
+ \put(0.01370736,0.19722543){\color[rgb]{0,0,0}\makebox(0,0)[lt]{\begin{minipage}{0.15987467\unitlength}\raggedright $\mu$\end{minipage}}}%
+ \put(0,0){\includegraphics[width=\unitlength,page=2]{torus_mu_lambda.pdf}}%
+ \put(0.80750641,0.20322471){\color[rgb]{0,0,0}\makebox(0,0)[lt]{\begin{minipage}{0.14685617\unitlength}\raggedright $K$\end{minipage}}}%
+ \put(0,0){\includegraphics[width=\unitlength,page=3]{torus_mu_lambda.pdf}}%
+ \put(0.66285955,0.08408629){\color[rgb]{0,0,0}\makebox(0,0)[lt]{\begin{minipage}{0.38612036\unitlength}\raggedright $N(K) = D^2 \times S^1$\end{minipage}}}%
+ \end{picture}%
+\endgroup%
diff --git a/images/torus_mu_lambda.svg b/images/torus_mu_lambda.svg
new file mode 100644
index 0000000..79eab56
--- /dev/null
+++ b/images/torus_mu_lambda.svg
@@ -0,0 +1,1109 @@
+
+
+
+
diff --git a/lec_03_06.tex b/lec_03_06.tex
index 669340b..5ae620f 100644
--- a/lec_03_06.tex
+++ b/lec_03_06.tex
@@ -69,5 +69,31 @@ S^1 \times \pt &\longrightarrow \pt \times S^1 \\
}
\end{figure}
+\begin{proof}[Sketch of proof]
+We will show that each diffeomorphism is isotopic to $\begin{pmatrix}
+p & q\\
+r & s
+\end{pmatrix}$.
+\begin{equation*}
+\quot{\Diff_+(S^1 \times S^1)}{\Iso(S^1 \times S^1)} = \mcg(S^1 \times S^1) = \Sl(2, \mathbb{Z})
+\end{equation*}
+\begin{figure}[h]
+\fontsize{20}{10}\selectfont
+\centering{
+\def\svgwidth{\linewidth}
+\resizebox{0.4\textwidth}{!}{\input{images/torus_mu_lambda.pdf_tex}}
+\caption{Choice of meridian and longitude.}
+\label{fig:torus_twist}
+}
+\end{figure}
+\end{proof}
+Let $N = D^2 \times S$ be a tubular neighbourhood of a knot $K$. Consider its boundary $\partial N = S^1 \times S^1$. There exists a simple closed curve $\mu \subset \partial N$ (a meridian) that bounds a disk in $N$. We choose another simple closed curve $\lambda$ (a longitude) so that $\Lk(\lambda, K) = 0$. \\
+????????
+\\
+$\lambda \mu = 1 $ intersection\\
+$\pi_0 (\Gl(2, \mathbb{R})$\\
+???????????
+\\
+In other words a homotopy class: $[\lambda] = 0$ in $H_1(S^3 \setminus N, \mathbb{Z})$.
diff --git a/lec_04_03.tex b/lec_04_03.tex
index a76580f..eeb7dda 100644
--- a/lec_04_03.tex
+++ b/lec_04_03.tex
@@ -1,4 +1,4 @@
-\subsection{Existence of Seifert surface - second proof}
+\subsection{Existence of a Seifert surface - second proof}
%\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}
@@ -209,7 +209,8 @@ There are not trivial knots with Alexander polynomial equal $1$, for example:
$\Delta_{11n34} \equiv 1$.
\end{example}
-\subsection{Decomposition of $3$-sphere}
+\subsection{Decomposition of \texorpdfstring{
+$3$-sphere}{3-sphere}}
We know that $3$ - sphere can be obtained by gluing two solid tori:
\[
S^3 = \partial D^4 = \partial (D^2 \times D^2) = (D^2 \times S^1) \cup (S^1 \times D^2).
diff --git a/lec_06_05.tex b/lec_06_05.tex
index 8af4b6a..601c6b4 100644
--- a/lec_06_05.tex
+++ b/lec_06_05.tex
@@ -17,12 +17,17 @@ The infinite cyclic cover of a knot complement $X$ is the cover associated with
\label{fig:covering}
}
\end{figure}
+\noindent
+\subsection{Double branched cover.}
+Let $K \subset S^3$ be a knot and $\Sigma$
+its Seifert surface.
+Let us consider a knot complement $S^3 \setminus N(K)$.
\begin{figure}[h]
\fontsize{10}{10}\selectfont
\centering{
\def\svgwidth{\linewidth}
\resizebox{0.8\textwidth}{!}{\input{images/knot_complement.pdf_tex}}
-\caption{A knot complement.}
+\caption{The double cover of the $3$-sphere branched over a knot $K$.}
\label{fig:complement}
}
\end{figure}
diff --git a/lec_08_04.tex b/lec_08_04.tex
index e432602..4dce8d2 100644
--- a/lec_08_04.tex
+++ b/lec_08_04.tex
@@ -1,9 +1,9 @@
-$X$ is a closed orientable four-manifold. Assume $\pi_1(X) = 0$ (it is not needed to define the intersection form). In particular $H_1(X) = 0$.
+$X$ is a closed orientable four-manifold. For simplicity assume $\pi_1(X) = 0$ (it is not needed to define the intersection form). In particular $H_1(X) = 0$.
$H_2$ is free (exercise).
-\begin{align*}
-H_2(X, \mathbb{Z}) \xrightarrow{\text{Poincar\'e duality}} H^2(X, \mathbb{Z} ) \xrightarrow{\text{evaluation}}\Hom(H_2(X, \mathbb{Z}), \mathbb{Z})
-\end{align*}
+\[
+H_2(X, \mathbb{Z}) \xrightarrow{\text{Poincar\'e duality}} H^2(X, \mathbb{Z} ) \xrightarrow{\text{evaluation}}\Hom(H_2(X, \mathbb{Z}), \mathbb{Z}).
+\]
\noindent
Intersection form:
$H_2(X, \mathbb{Z}) \times
@@ -18,7 +18,7 @@ Let $A$ and $B$ be closed, oriented surfaces in $X$.
\resizebox{0.5\textwidth}{!}{\input{images/intersection_form_A_B.pdf_tex}}
}
\caption{$T_X A + T_X B = T_X X$
-}\label{fig:torus_alpha_beta}
+}\label{fig:intersection}
\end{figure}
???????????????????????
\begin{align*}
diff --git a/lec_10_06.tex b/lec_10_06.tex
index 9541a40..1b1f8a4 100644
--- a/lec_10_06.tex
+++ b/lec_10_06.tex
@@ -1,48 +1 @@
-
-
-
-
-\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}
+Consider a surgery
diff --git a/lec_20_05.tex b/lec_20_05.tex
index 020911b..70cf05d 100644
--- a/lec_20_05.tex
+++ b/lec_20_05.tex
@@ -60,7 +60,7 @@ H_1(\widetilde{X}, \mathbb{Z}[t, t^{-1}]) \longrightarrow \quot{\mathbb{Q}}{\mat
\begin{align*}
H_1(\widetilde{X}, \mathbb{Z}[t, t^{-1}]) \times
H_1(\widetilde{X}, \mathbb{Z}[t, t^{-1}]) &\longrightarrow \quot{\mathbb{Q}}{\mathbb{Z}[t, t^{-1}]}\\
-(\alpha, \beta) \quad &\mapsto \alpha^{-1}(t -1)(tV - V^T)^{-1}\beta
+(\alpha, \beta) \quad &\mapsto \alpha^{-1}{(t -1)(tV - V^T)}^{-1}\beta
\end{align*}
\end{fact}
\noindent
diff --git a/lec_25_02.tex b/lec_25_02.tex
index 7586384..3e1c85b 100644
--- a/lec_25_02.tex
+++ b/lec_25_02.tex
@@ -6,8 +6,24 @@ A knot $K$ in $S^3$ is a smooth (PL - smooth) embedding of a circle $S^1$ in $S^
\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}
+\begin{figure}[h]
+\includegraphics[width=0.08\textwidth]
+{unknot.png}
+\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}
+\end{figure}
+
+
+
\begin{itemize}
\item
Knots:
@@ -41,12 +57,15 @@ Two knots $K_0$ and $K_1$ are isotopic if and only if they are ambient isotopic,
& \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 $S^1$.
\end{definition}
+
\begin{definition}
A link with k - components is a (smooth) embedding of $\overbrace{S^1 \sqcup \ldots \sqcup S^1}^k$ in $S^3$.
\end{definition}
+
\begin{example}
Links:
\begin{itemize}
diff --git a/lec_27_05.tex b/lec_27_05.tex
index 1b7b240..a18c596 100644
--- a/lec_27_05.tex
+++ b/lec_27_05.tex
@@ -62,7 +62,80 @@ H_1(\Sigma(K), \mathbb{Z})
(a, b) \mapsto a{(V + V^T)}^{-1} b
\end{eqnarray*}
???????????????????\\
+???????????????????\\
\begin{eqnarray*}
y \mapsto y + Az \\
-\overline{x^T} A^{-1}(y + Az) = \overline{x^T} A^{-1} + \overline{x^T} \mathbb{1} z
+\overline{x^T} A^{-1}(y + Az) = \overline{x^T} A^{-1} y + \overline{x^T} \mathbb{1} z = \overline{x^T} A^{-1}y \in \quot{\Omega}{\Lambda} \\
+\overline{x^T} \mathbb{1} z \in \Lambda \\
+H_1(\widetilde{X}, \Lambda) =
+\quot{ \Lambda^n }{(Vt - V) \Lambda^n}
+\\
+(a, b) \mapsto \overline{a^T}(Vt - V^T)^{-1} (t -1)b
\end{eqnarray*}
+(Blanchfield '59)
+\begin{theorem}[Kearton '75, Friedl, Powell '15]
+There exits a matrix $A$ representing the Blanchfield paring over $\mathbb{Z}[t, t^{-1}]$. The size of $A$ is a size of Seifert form.
+\end{theorem}
+Remark:
+\begin{enumerate}
+\item
+Over $\mathbb{R}$ we can take $A$ to be diagonal.
+\item
+The jump of signature function at $\xi$ is
+ equal to
+ \[
+ \lim_{t \rightarrow 0^+} \sign A (e^{it} \xi) - \sign A(e^{-it} \xi).
+ \]
+ \item
+ The minimal size of a matrix $A$ that presents a Blanchfield paring (over $\mathbb{Z}[t, t^{-1}]$) for a knot $K$ is a knot invariant.
+\end{enumerate}
+
+\subsection{The unknotting number}
+Let $K$ be a knot and $D$ a knot diagram. A crossing change is a modification of a knot diagram by one of following changes
+\begin{align*}
+\PICorientpluscross \mapsto \PICorientminuscross ,\\
+\PICorientminuscross \mapsto\PICorientminuscross.
+\end{align*}
+The unknotting number $u(K)$ is a number of crossing changes needed to turn a knot into an unknot, where the minimum is taken over all diagrams of a given knot.
+\begin{definition}
+A Gordian distance $G(K, K^\prime)$ between knots $K$ and $K^\prime$ is the minimal number of crossing changes required to turn $K$ into $K^\prime$.
+\end{definition}
+\begin{problem}
+Prove that:
+\[
+G(K, K^{\prime\prime})
+\leq
+G(K, K^{\prime})
++
+G(K^\prime, K^{\prime\prime}).
+\]
+Open problem:
+\[
+u(K\# K^\prime) = u(K) + u(K^\prime).
+\]
+\end{problem}
+\begin{lemma}[Scharlemann '84]
+Unknotting number one knots are prime.
+\end{lemma}
+\subsection*{Tools to bound unknotting number}
+\begin{theorem}
+For any symmetric polynomial $\Delta \in \mathbb{Z}[t, t^{-1}]$ such that $\Delta(1) = 1$, there exists a knot $K$ such that:
+\begin{enumerate}
+\item
+$K$ has unknotting number $1$,
+\item
+$\Delta_K = \Delta$.
+\end{enumerate}
+\end{theorem}
+Let us consider a knot $K$ and its Seifert surface $\Sigma$.
+
+the Seifert form for $K_-$
+\\
+the Seifert form for $K_+$
+\\
+$S_- + S_+$ differs from
+by a term in the bottom right corner
+
+Let $A$ be a symmetric $n \times n$ matrix over $\mathbb{R}$. Let $A_1, \dots, A_n$ be minors of $A$. \\
+Let $\epsilon_0 = 1$
+If
\ No newline at end of file
diff --git a/lectures_on_knot_theory.tex b/lectures_on_knot_theory.tex
index be651af..097d940 100644
--- a/lectures_on_knot_theory.tex
+++ b/lectures_on_knot_theory.tex
@@ -98,6 +98,10 @@
\DeclareMathOperator{\sign}{sign}
\DeclareMathOperator{\odd}{odd}
\DeclareMathOperator{\even}{even}
+\DeclareMathOperator{\Diff}{Diff}
+\DeclareMathOperator{\Iso}{Iso}
+\DeclareMathOperator{\mcg}{MCG}
+
@@ -126,45 +130,92 @@
%\newpage
%\input{myNotes}
-\section{Basic definitions \hfill\DTMdate{2019-02-25}}
+\section{Basic definitions
+\texorpdfstring{
+\hfill \DTMdate{2019-02-25}}
+{}}
\input{lec_25_02.tex}
-\section{Alexander polynomial \hfill\DTMdate{2019-03-04}}
+\section{Alexander polynomial
+\texorpdfstring{
+\hfill\DTMdate{2019-03-04}}
+{}}
\input{lec_04_03.tex}
%add Hurewicz theorem?
\section{Examples of knot classes
+\texorpdfstring{
\hfill\DTMdate{2019-03-11}}
+{}}
\input{lec_11_03.tex}
-\section{Concordance group \hfill\DTMdate{2019-03-18}}
+\section{Concordance group
+\texorpdfstring{
+\hfill\DTMdate{2019-03-18}}
+{}}
\input{lec_18_03.tex}
-\section{Genus $g$ cobordism \hfill\DTMdate{2019-03-25}}
+\section{Genus
+\texorpdfstring{$g$}{g}
+cobordism
+\texorpdfstring{
+\hfill\DTMdate{2019-03-25}}
+{}}
\input{lec_25_03.tex}
-\section{\hfill\DTMdate{2019-04-08}}
+\section{
+\texorpdfstring{
+\hfill\DTMdate{2019-04-08}}
+{}}
\input{lec_08_04.tex}
-\section{Linking form\hfill\DTMdate{2019-04-15}}
+\section{Linking form
+\texorpdfstring{
+\hfill\DTMdate{2019-04-15}}
+{}}
\input{lec_15_04.tex}
-\section{\hfill\DTMdate{2019-05-06}}
+\section{
+\texorpdfstring{
+\hfill\DTMdate{2019-05-06}}
+{}}
\input{lec_06_05.tex}
-\section{\hfill\DTMdate{2019-05-20}}
+% no lecture at 13.05
+%\section{\hfill\DTMdate{2019-05-20}}
+%\input{lec_13_05.tex}
+
+\section{
+\texorpdfstring{
+\hfill\DTMdate{2019-05-20}}
+{}}
\input{lec_20_05.tex}
-\section{\hfill\DTMdate{2019-05-27}}
+\section{
+\texorpdfstring{
+\hfill\DTMdate{2019-05-27}}
+{}}
\input{lec_27_05.tex}
-\section{Surgery \hfill\DTMdate{2019-06-03}}
+\section{
+\texorpdfstring{
+Surgery \hfill\DTMdate{2019-06-03}}
+{}}
\input{lec_03_06.tex}
-\section{Surgery\hfill\DTMdate{2019-06-03}}
+\section{Surgery
+\texorpdfstring{
+\hfill\DTMdate{2019-06-10}}
+{}}
\input{lec_10_06.tex}
+\section{Mess
+\texorpdfstring{
+\hfill\DTMdate{2019-06-17}}
+{}}
+\input{mess.tex}
+
\end{document}