303 lines
10 KiB
TeX
303 lines
10 KiB
TeX
\begin{definition}
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A knot $K$ in $S^3$ is a smooth (PL - smooth) embedding of a circle $S^1$ in $S^3$:
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\begin{align*}
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\varphi: S^1 \hookrightarrow S^3
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\end{align*}
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\end{definition}
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\noindent
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Usually we think about a knot as an image of an embedding: $K = \varphi(S^1)$.
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Some basic examples and counterexamples are shown respectively in \autoref{fig:unknot} and \autoref{fig:notknot}.
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\begin{example}
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\begin{figure}[h]
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\includegraphics[width=0.08\textwidth]
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{unknot.png}
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\caption{Knots examples: unknot (left) and trefoil (right).}
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\label{fig:unknot}
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\end{figure}
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\begin{figure}[h]
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\includegraphics[width=0.08\textwidth]
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{unknot.png}
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\caption{Knots examples: unknot (left) and trefoil (right).}
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\label{fig:notknot}
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\end{figure}
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\begin{itemize}
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\item
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Knots:
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\includegraphics[width=0.08\textwidth]{unknot.png} (unknot),
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\includegraphics[width=0.08\textwidth]{trefoil.png} (trefoil).
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\item
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Not knots:
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\includegraphics[width=0.12\textwidth]{not_injective_knot.png}
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(it is not an injection),
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\includegraphics[width=0.08\textwidth]{not_smooth_knot.png}
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(it is not smooth).
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\end{itemize}
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\end{example}
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\begin{definition}
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%\hfill\\
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Two knots $K_0 = \varphi_0(S^1)$, $K_1 = \varphi_1(S^1)$ are equivalent if the embeddings $\varphi_0$ and $\varphi_1$ are isotopic, that is there exists a continues function
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\begin{align*}
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&\Phi: S^1 \times [0, 1] \hookrightarrow S^3, \\
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&\Phi(x, t) = \Phi_t(x)
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\end{align*}
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such that $\Phi_t$ is an embedding for any $t \in [0,1]$, $\Phi_0 = \varphi_0$ and
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$\Phi_1 = \varphi_1$.
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\end{definition}
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\begin{theorem}
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Two knots $K_0$ and $K_1$ are isotopic if and only if they are ambient isotopic, i.e. there exists a family of self-diffeomorphisms $\Psi = \{\psi_t: t \in [0, 1]\}$ such that:
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\begin{align*}
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&\psi(t) = \psi_t \text{ is continius on $t\in [0,1]$},\\
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&\psi_t: S^3 \hookrightarrow S^3,\\
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& \psi_0 = id ,\\
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& \psi_1(K_0) = K_1.
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\end{align*}
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\end{theorem}
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\begin{definition}
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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 $S^1$.
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\end{definition}
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\begin{definition}
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A link with k - components is a (smooth) embedding of $\overbrace{S^1 \sqcup \ldots \sqcup S^1}^k$ in $S^3$.
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\end{definition}
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\begin{example}
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Links:
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\begin{itemize}
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\item
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a trivial link with $3$ components:
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\includegraphics[width=0.2\textwidth]{3unknots.png},
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\item
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a Hopf link: \includegraphics[width=0.13\textwidth]{Hopf.png},
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\item
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a Whitehead link:
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\includegraphics[width=0.13\textwidth]{WhiteheadLink.png},
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\item
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a Borromean link:
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\includegraphics[width=0.1\textwidth]{BorromeanRings.png}.
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\end{itemize}
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\end{example}
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%
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%
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%
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\begin{definition}
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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:
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\begin{enumerate}[label={(\arabic*)}]
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\item
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$D_{\pi |_L}$ is non degenerate: \includegraphics[width=0.05\textwidth]{LinkDiagram1.png},
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\item the double points are not degenerate: \includegraphics[width=0.03\textwidth]{LinkDiagram2.png},
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\item there are no triple point: \includegraphics[width=0.05\textwidth]{LinkDiagram3.png}.
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\end{enumerate}
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\end{definition}
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\noindent
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There are under- and overcrossings (tunnels and bridges) on a link diagrams with an obvious meaning.
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\begin{fact}
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Every link admits a link diagram.
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\end{fact}
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\noindent
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Let $D$ be a diagram of an oriented link (to each component of a link we add an arrow in the diagram).
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We can distinguish two types of crossings: right-handed
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$\left(\PICorientpluscross\right)$, called a positive crossing, and left-handed $\left(\PICorientminuscross\right)$, called a negative crossing.
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\subsection{Reidemeister moves}
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A Reidemeister move is one of the three types of operation on a link diagram as shown below:
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\begin{enumerate}[label=\Roman*]
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\item\hfill\\
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\includegraphics[width=0.6\textwidth]{rm1.png},
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\item\hfill\\\includegraphics[width=0.6\textwidth]{rm2.png},
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\item\hfill\\\includegraphics[width=0.4\textwidth]{rm3.png}.
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\end{enumerate}
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\begin{theorem} [Reidemeister, 1927 ]
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Two diagrams of the same link can be
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deformed into each other by a finite sequence of Reidemeister moves (and isotopy of the plane).
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\end{theorem}
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%
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%
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%
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%The number of Reidemeister Moves Needed for Unknotting
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%Joel Hass, Jeffrey C. Lagarias
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%(Submitted on 2 Jul 1998)
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% Piotr Sumata, praca magisterska
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% proof - transversality theorem (Thom)
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%Singularities of Differentiable Maps
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%Authors: Arnold, V.I., Varchenko, Alexander, Gusein-Zade, S.M.
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\subsection{Seifert surface}
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\noindent
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Let $D$ be an oriented diagram of a link $L$. We change the diagram by smoothing each crossing:
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\begin{align*}
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\PICorientpluscross \mapsto \PICorientLRsplit,\\
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\PICorientminuscross \mapsto \PICorientLRsplit.
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\end{align*}
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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 $\partial \Sigma = L$.\\
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\begin{figure}[h]
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\fontsize{15}{10}\selectfont
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\centering{
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\def\svgwidth{\linewidth}
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\resizebox{0.8\textwidth}{!}{\input{images/seifert_alg.pdf_tex}}
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\caption{Constructing a Seifert surface.}
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\label{fig:SeifertAlg}
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}
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\end{figure}
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\noindent
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Note: the obtained surface isn't unique and in general 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 $D_2$. Then we glue both components on the boundaries: $\partial D_1$ and $\partial D_2$.
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\begin{figure}[h]
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\begin{center}
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\includegraphics[width=0.6\textwidth]{seifert_connect.png}
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\end{center}
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\caption{Connecting two surfaces.}
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\label{fig:SeifertConnect}
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\end{figure}
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\begin{theorem}[Seifert]
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\label{theo:Seifert}
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Every link in $S^3$ bounds a surface $\Sigma$ that is compact, connected and orientable. Such a surface is called a Seifert surface.
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\end{theorem}
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%
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\begin{figure}[h]
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\fontsize{12}{10}\selectfont
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\centering{
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\def\svgwidth{\linewidth}
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\resizebox{1\textwidth}{!}{\input{images/torus_1_2_3.pdf_tex}}
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\caption{Genus of an orientable surface.}
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\label{fig:genera}
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}
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\end{figure}
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%
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%
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\begin{definition}
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The three genus $g_3(K)$ ($g(K)$) of a knot $K$ is the minimal genus of a Seifert surface $\Sigma$ for $K$.
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\end{definition}
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\begin{corollary}
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A knot $K$ is trivial if and only $g_3(K) = 0$.
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\end{corollary}
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\noindent
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Remark: there are knots that admit non isotopic Seifert surfaces of minimal genus (András Juhász, 2008).
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\begin{definition}
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Suppose $\alpha$ and $\beta$ are two simple closed curves in $\mathbb{R}^3$.
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On a diagram $L$ consider all crossings between $\alpha$ and $\beta$. Let $N_+$ be the number of positive crossings, $N_-$ - negative. Then the linking number: $\Lk(\alpha, \beta) = \frac{1}{2}(N_+ - N_-)$.
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\end{definition}
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\begin{definition}
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\label{def:lk_via_homo}
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Let $\alpha$ and $\beta$ be two disjoint simple cross curves in $S^3$.
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Let $\nu(\beta)$ be a tubular neighbourhood of $\beta$. The linking number can be interpreted via first homology group, where $\Lk(\alpha, \beta)$ is equal to evaluation of $\alpha$ as element of first homology group of the complement of $\beta$:
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\[
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\alpha \in H_1(S^3 \setminus \nu(\beta), \mathbb{Z}) \cong \mathbb{Z}.\]
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\end{definition}
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\begin{example}
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\begin{itemize}
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\item
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A Hopf link:
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\begin{figure}[h]
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\fontsize{20}{10}\selectfont
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\centering{
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\def\svgwidth{\linewidth}
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\resizebox{0.4\textwidth}{!}{\input{images/linking_hopf.pdf_tex}},
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}
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\end{figure}
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\item
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$T(6, 2)$ link:
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\begin{figure}[h]
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\fontsize{20}{10}\selectfont
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\centering{
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\def\svgwidth{\linewidth}
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\resizebox{0.4\textwidth}{!}{\input{images/linking_torus_6_2.pdf_tex}}.
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}
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\end{figure}
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\end{itemize}
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\end{example}
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\begin{fact}
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$
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g_3(\Sigma) = \frac{1}{2} b_1 (\Sigma) =
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\frac{1}{2} \dim_{\mathbb{R}}H_1(\Sigma, \mathbb{R}),
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$
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where $b_1$ is first Betti number of $\Sigma$.
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\end{fact}
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\subsection{Seifert matrix}
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Let $L$ be a link and $\Sigma$ be an oriented Seifert surface for $L$. Choose a basis for $H_1(\Sigma, \mathbb{Z})$ consisting of simple closed curves $\alpha_1, \dots, \alpha_n$.
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Let $\alpha_1^+, \dots \alpha_n^+$ be copies of $\alpha_i$ lifted up off the surface (push up along a vector field normal to $\Sigma$). Note that elements $\alpha_i$ are contained in the Seifert surface while all $\alpha_i^+$ don't intersect the surface.
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Let $\Lk(\alpha_i, \alpha_j^+) = \{a_{ij}\}$. Then the matrix $S = \{a_{ij}\}_{i, j =1}^n$ is called a Seifert matrix for $L$. Note that by choosing a different basis we get a different matrix.
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\begin{figure}[h]
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\fontsize{20}{10}\selectfont
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\centering{
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\def\svgwidth{\linewidth}
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\resizebox{0.8\textwidth}{!}{\input{images/seifert_matrix.pdf_tex}}
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}
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\end{figure}
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\begin{theorem}
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The Seifert matrices $S_1$ and $S_2$ for the same link $L$ are S-equivalent, that is, $S_2$ can be obtained from $S_1$ by a sequence of following moves:
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\begin{enumerate}[label={(\arabic*)}]
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\item
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$V \rightarrow AVA^T$, where $A$ is a matrix with integer coefficients,
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\item
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$V \rightarrow
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\begin{pmatrix}
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\begin{array}{c|c}
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V &
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\begin{matrix}
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\ast & 0 \\
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\sdots & \sdots\\
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\ast & 0
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\end{matrix} \\
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\hline
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\begin{matrix}
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\ast & \dots & \ast\\
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0 & \dots & 0
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\end{matrix}
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&
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\begin{matrix}
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0 & 0\\
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1 & 0
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\end{matrix}
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\end{array}
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\end{pmatrix} \quad$
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or
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$\quad
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V \rightarrow
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\begin{pmatrix}
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\begin{array}{c|c}
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V &
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\begin{matrix}
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\ast & 0 \\
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\sdots & \sdots\\
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\ast & 0
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\end{matrix} \\
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\hline
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\begin{matrix}
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\ast & \dots & \ast\\
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0 & \dots & 0
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\end{matrix}
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&
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\begin{matrix}
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0 & 1\\
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0 & 0
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\end{matrix}
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\end{array}
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\end{pmatrix},$
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\item
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inverse of (2).
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\end{enumerate}
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\end{theorem}
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