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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}