diff --git a/images/WhiteheadLink.png b/images/WhiteheadLink.png
index 646e877..454d4f0 100644
Binary files a/images/WhiteheadLink.png and b/images/WhiteheadLink.png differ
diff --git a/images/ball_complement_copies.svg b/images/ball_complement_copies.svg
new file mode 100644
index 0000000..c28657b
--- /dev/null
+++ b/images/ball_complement_copies.svg
@@ -0,0 +1,860 @@
+
+
+
+
diff --git a/images/drawing.svg b/images/drawing.svg
new file mode 100644
index 0000000..a1ba670
--- /dev/null
+++ b/images/drawing.svg
@@ -0,0 +1,101 @@
+
+
+
+
diff --git a/images/not_knots.svg b/images/not_knots.svg
index b4550fc..a796095 100644
--- a/images/not_knots.svg
+++ b/images/not_knots.svg
@@ -400,7 +400,7 @@
inkscape:pageopacity="0.0"
inkscape:pageshadow="2"
inkscape:zoom="0.91545961"
- inkscape:cx="911.38483"
+ inkscape:cx="925.71075"
inkscape:cy="1195.7783"
inkscape:document-units="px"
inkscape:current-layer="layer1"
@@ -700,8 +700,35 @@
inkscape:export-ydpi="90" />
+ inkscape:connector-curvature="0"
+ sodipodi:nodetypes="cc"
+ inkscape:export-xdpi="90"
+ inkscape:export-ydpi="90" />
+
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diff --git a/images/rm.svg b/images/rm.svg
new file mode 100644
index 0000000..63b89eb
--- /dev/null
+++ b/images/rm.svg
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new file mode 100644
index 0000000..f7c0c45
--- /dev/null
+++ b/images/rm.svg.2019_05_30_02_39_32.0.svg
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new file mode 100644
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diff --git a/images/rm2.png b/images/rm2.png
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diff --git a/images/rm3.png b/images/rm3.png
new file mode 100644
index 0000000..eae5bbb
Binary files /dev/null and b/images/rm3.png differ
diff --git a/images/seifert.pdf b/images/seifert.pdf
new file mode 100644
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Binary files /dev/null and b/images/seifert.pdf differ
diff --git a/images/seifert.pdf.2019_05_31_08_05_24.0.svg b/images/seifert.pdf.2019_05_31_08_05_24.0.svg
new file mode 100644
index 0000000..c44d3a7
--- /dev/null
+++ b/images/seifert.pdf.2019_05_31_08_05_24.0.svg
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diff --git a/images/seifert.pdf.2019_05_31_08_32_31.0.svg b/images/seifert.pdf.2019_05_31_08_32_31.0.svg
new file mode 100644
index 0000000..78b8613
--- /dev/null
+++ b/images/seifert.pdf.2019_05_31_08_32_31.0.svg
@@ -0,0 +1,2278 @@
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diff --git a/images/seifert.pdf_tex b/images/seifert.pdf_tex
new file mode 100644
index 0000000..2c1611e
--- /dev/null
+++ b/images/seifert.pdf_tex
@@ -0,0 +1,81 @@
+%% Creator: Inkscape inkscape 0.91, www.inkscape.org
+%% PDF/EPS/PS + LaTeX output extension by Johan Engelen, 2010
+%% Accompanies image file 'seifert.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}{773.77219373bp}%
+ \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}(1,0.34116869)%
+ \put(0,0){\includegraphics[width=\unitlength,page=1]{seifert.pdf}}%
+ \put(1.48606212,-0.64968059){\color[rgb]{0,0,0}\makebox(0,0)[lt]{\begin{minipage}{0.59339172\unitlength}\raggedright $B^4$\\ \end{minipage}}}%
+ \put(0.88568904,1.12362657){\color[rgb]{0,0,0}\makebox(0,0)[lt]{\begin{minipage}{0.6133154\unitlength}\raggedright \end{minipage}}}%
+ \put(0,0){\includegraphics[width=\unitlength,page=2]{seifert.pdf}}%
+ \put(1.48380337,-0.3018334){\color[rgb]{0,0,0}\makebox(0,0)[lt]{\begin{minipage}{0.59339172\unitlength}\raggedright $B^4$\\ \end{minipage}}}%
+ \put(0,0){\includegraphics[width=\unitlength,page=3]{seifert.pdf}}%
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diff --git a/images/seifert3d b/images/seifert3d
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diff --git a/images/seifert3d.png b/images/seifert3d.png
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Binary files /dev/null and b/images/seifert3d.png differ
diff --git a/images/seifert_bands.pdf b/images/seifert_bands.pdf
new file mode 100644
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Binary files /dev/null and b/images/seifert_bands.pdf differ
diff --git a/images/seifert_bands.pdf_tex b/images/seifert_bands.pdf_tex
new file mode 100644
index 0000000..2aebd53
--- /dev/null
+++ b/images/seifert_bands.pdf_tex
@@ -0,0 +1,75 @@
+%% Creator: Inkscape inkscape 0.91, www.inkscape.org
+%% PDF/EPS/PS + LaTeX output extension by Johan Engelen, 2010
+%% Accompanies image file 'seifert_bands.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]{}%
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+ \ifx\svgscale\undefined%
+ \relax%
+ \else%
+ \setlength{\unitlength}{\unitlength * \real{\svgscale}}%
+ \fi%
+ \else%
+ \setlength{\unitlength}{\svgwidth}%
+ \fi%
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+ \makeatother%
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+ \put(0.22955492,1.21708586){\color[rgb]{0,0,0}\makebox(0,0)[lt]{\begin{minipage}{0.55033279\unitlength}\raggedright \end{minipage}}}%
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diff --git a/images/seifert_bands.svg b/images/seifert_bands.svg
new file mode 100644
index 0000000..c79e5b4
--- /dev/null
+++ b/images/seifert_bands.svg
@@ -0,0 +1,2693 @@
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+
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diff --git a/images/seifert_surface.pdf b/images/seifert_surface.pdf
new file mode 100644
index 0000000..69fa214
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diff --git a/images/seifert_surface.pdf_tex b/images/seifert_surface.pdf_tex
new file mode 100644
index 0000000..214c791
--- /dev/null
+++ b/images/seifert_surface.pdf_tex
@@ -0,0 +1,89 @@
+%% Creator: Inkscape inkscape 0.91, www.inkscape.org
+%% PDF/EPS/PS + LaTeX output extension by Johan Engelen, 2010
+%% Accompanies image file 'seifert_surface.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][]{}%
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+ \errmessage{(Inkscape) Transparency is used (non-zero) for the text in Inkscape, but the package 'transparent.sty' is not loaded}%
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new file mode 100644
index 0000000..b330211
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+
+
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diff --git a/lectures_on_knot_theory.tex b/lectures_on_knot_theory.tex
index 10e44b2..fbb9723 100644
--- a/lectures_on_knot_theory.tex
+++ b/lectures_on_knot_theory.tex
@@ -155,35 +155,67 @@ Borromean link:
%
%
\begin{definition}
-A link diagram is a picture over projection of a link is $S^3$($\mathbb{R}^3$) such that:
+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
${D_{\pi}}_{\big|L}$ is non degenerate: \includegraphics[width=0.05\textwidth]{LinkDiagram1.png},
-\item the double points are not degenerate: \includegraphics[width=0.05\textwidth]{LinkDiagram2.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}
\end{definition}
+\noindent
There are under- and overcrossings (tunnels and bridges) on a link diagrams with an obvious meaning.\\
Every link admits a link diagram.
-%\begin{comment}
+\\
+Let $D$ be a diagram of an oriented link (to each component of a link we add an arrow in the diagram).\\
+We can distinguish two types of crossings: right-handed
+$\left(\PICorientpluscross\right)$, called a positive crossing, and left-handed $\left(\PICorientminuscross\right)$, called a negative crossing.
\section*{Reidemeister moves}
-A Reidemeister move is one of the three types of operation on a link diagram as shown in Figure~\ref{fig: reidemeister}.
-%
-The first Reidemeister move inserts or removes a coil.
-%
-The second Reidemeister move slides a strand and inserts or removes two crossings of opposite sign.
-%
-The third Reidemeister move slides a strand over or under a crossing.
-\begin{figure}[H]
-\centering
-\includegraphics[width=0.7\textwidth]{moves.png}
-\caption{\label{fig: reidemeister}Reidemeister moves (adapted from Adams).}
-\end{figure}
-\begin{theorem} [Reidemeister’s Theorem]
+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}.
+\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).
\end{theorem}
+%
+%
+%
+%The number of Reidemeister Moves Needed for Unknotting
+%Joel Hass, Jeffrey C. Lagarias
+%(Submitted on 2 Jul 1998)
+\subsection*{Seifert surface}
+\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
+\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 $\partial \Sigma = L$.\\
+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]
+\fontsize{15}{10}\selectfont
+\centering{
+\def\svgwidth{\linewidth}
+\resizebox{0.7\textwidth}{!}{\input{images/seifert_surface.pdf_tex}}
+\caption{Constructing a Seifert surface.}
+\label{fig:surfaceSeifert}
+}
+\end{figure}
+
+\includegraphics[width=0.3\textwidth]{seifert3d.png},
+
+ % transversality theorem
+%Thom ?
+%Singularities of Differentiable Maps
+%Authors: Arnold, V.I., Varchenko, Alexander, Gusein-Zade, S.M.
\section{}
\begin{example}