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some inkspace pictures

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Maria Marchwicka 2019-05-31 20:33:27 +02:00
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64
lectures_on_knot_theory.tex
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@ -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}