some inkspace pictures
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Maria Marchwicka
parent
d62b96a337
commit
8574646027
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Before Width: | Height: | Size: 57 KiB After Width: | Height: | Size: 10 KiB |
860
images/ball_complement_copies.svg
Normal file
860
images/ball_complement_copies.svg
Normal file
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%% Creator: Inkscape inkscape 0.91, www.inkscape.org
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%% PDF/EPS/PS + LaTeX output extension by Johan Engelen, 2010
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%% \import{<path to file>}{<filename>.pdf_tex}
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%% Alternatively, one can specify
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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}
|
||||
|
||||
Loading…
Reference in New Issue
Block a user