lectures_on_knot_theory/lec_25_02.tex

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\begin{definition}
A knot $K$ in $S^3$ is a smooth (PL - smooth)
embedding of a circle $S^1$ in $S^3$:
\[
\varphi: S^1 \hookrightarrow S^3
\]
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\end{definition}
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\noindent
Usually we think about a knot
as an image of an embedding:
$K = \varphi(S^1)$.
Some basic examples and counterexamples
are shown respectively in
\autoref{fig:unknot} and
\autoref{fig:notknot}.
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\begin{figure}[h]
\centering
\begin{subfigure}{0.3\textwidth}
\centering
\includegraphics[width=0.5\textwidth]
{unknot.png}
\end{subfigure}
\begin{subfigure}{0.3\textwidth}
\centering
\includegraphics[width=0.5\textwidth]
{trefoil.png}
\end{subfigure}
\caption{Knots examples:
unknot (left) and trefoil (right).}
\label{fig:unknot}
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\end{figure}
\begin{figure}[h]
\centering
\begin{subfigure}{0.3\textwidth}
\centering
\includegraphics[width=0.5\textwidth]
{not_injective_knot.png}
\end{subfigure}
\begin{subfigure}{0.3\textwidth}
\centering
\includegraphics[width=0.5\textwidth]
{not_smooth_knot.png}
\end{subfigure}
\caption{Not-knots examples:
an image of
a function ${S^1\longrightarrow S^3}$
that isn't injective (left) and
of a function
that isn't smooth (right).}
\label{fig:notknot}
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\end{figure}
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\begin{definition}
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
\begin{align*}
&\Phi: S^1 \times
[0, 1] \hookrightarrow S^3, \\
&\Phi(x, t) = \Phi_t(x)
\end{align*}
such that
$\Phi_t$ is an embedding
for any $t \in [0,1]$,
$\Phi_0 = \varphi_0$ and
$\Phi_1 = \varphi_1$.
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\end{definition}
\begin{theorem}
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:
\begin{align*}
&\psi(t) = \psi_t
\text{ is continius on
$t\in [0,1]$},\\
&\psi_t: S^3 \hookrightarrow S^3,\\
& \psi_0 = id ,\\
& \psi_1(K_0) = K_1.
\end{align*}
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\end{theorem}
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\begin{definition}
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}
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}
Links:
\begin{itemize}
\item
a trivial link with $3$ components:
\includegraphics[width=0.2\textwidth]{3unknots.png},
\item
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a Hopf link: \includegraphics[width=0.13\textwidth]{Hopf.png},
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\item
a Whitehead link:
\includegraphics[width=0.13\textwidth]{WhiteheadLink.png},
\item
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a Borromean link:
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\includegraphics[width=0.1\textwidth]{BorromeanRings.png}.
\end{itemize}
\end{example}
%
%
%
\begin{definition}\label{def:link_diagram}
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
$D_{\pi |_L}$ is non degenerate,
\item
the double points are not degenerate,
\item there are no triple point.
\end{enumerate}
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\end{definition}
\noindent
By \Cref{def:link_diagram} the following pictures can not be a part of a diagram:
\includegraphics[width=0.05\textwidth]{LinkDiagram1.png},
\includegraphics[width=0.03\textwidth]{LinkDiagram2.png},
\includegraphics[width=0.05\textwidth]{LinkDiagram3.png}.
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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.
\begin{lemma}
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Every link admits a link diagram.
\end{lemma}
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\noindent
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
$\left(\PICorientpluscross\right)$, called a positive crossing, and left-handed $\left(\PICorientminuscross\right)$, called a negative crossing.
\subsection{Reidemeister moves}
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}.
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\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).
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\end{theorem}
%
%
%
%The number of Reidemeister Moves Needed for Unknotting
%Joel Hass, Jeffrey C. Lagarias
%(Submitted on 2 Jul 1998)
% Piotr Sumata, praca magisterska
% proof - transversality theorem (Thom)
%Singularities of Differentiable Maps
%Authors: Arnold, V.I., Varchenko, Alexander, Gusein-Zade, S.M.
\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.
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\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 $\partial \Sigma = L$.\\
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\begin{figure}[h]
\fontsize{15}{10}\selectfont
\centering{
\def\svgwidth{\linewidth}
\resizebox{0.8\textwidth}{!}
{\input{images/seifert_alg.pdf_tex}}
\caption{Constructing a Seifert surface.}
\label{fig:SeifertAlg}
}
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\end{figure}
\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]
\centering
\includegraphics[width=0.6\textwidth]
{seifert_connect.png}
\caption{Connecting two surfaces.}
\label{fig:SeifertConnect}
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\end{figure}
\begin{theorem}[Seifert]\label{theo:Seifert}
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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\begin{figure}[h]
\fontsize{12}{10}\selectfont
\centering
\def\svgwidth{\linewidth}
\resizebox{1\textwidth}{!}{
\input{images/torus_1_2_3.pdf_tex}}
\caption{Genus of an orientable surface.}
\label{fig:genera}
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\end{figure}
%
%
\begin{definition}
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}
\begin{corollary}
A knot $K$ is trivial if and only
$g_3(K) = 0$.
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\end{corollary}
\noindent
Remark: there are knots that admit non isotopic
Seifert surfaces of minimal genus
(András Juhász, 2008).
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\begin{definition}
Suppose $\alpha$ and $\beta$ are two
simple closed curves in $\mathbb{R}^3$.
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}
\begin{definition}\label{def:lk_via_homo}
Let $\alpha$ and $\beta$ be
two disjoint simple closed curves in $S^3$.
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 $\beta$:
\[
\alpha \in H_1(S^3 \setminus
\nu(\beta), \mathbb{Z})
\cong \mathbb{Z}.
\]
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\end{definition}
\begin{figure}[h]
\fontsize{10}{8}\selectfont
\centering
\def\svgwidth{\linewidth}
\resizebox{\textwidth}{!}{
% \centering
\begin{subfigure}{0.3\textwidth}
\centering
\def\svgwidth{\linewidth}
\resizebox{1\textwidth}{!}{
\input{images/linking_torus_6_2.pdf_tex}
}
\end{subfigure}
\begin{subfigure}{0.3\textwidth}
\centering
\def\svgwidth{\linewidth}
\resizebox{1\textwidth}{!}{
\input{images/linking_hopf.pdf_tex}
}
\end{subfigure}
}
\vspace*{10mm}
\caption{
Linking number of a Hopf link (left)
and a torus link $T(6, 2)$ (right).
}
\label{fig:unknot}
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\end{figure}
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\begin{fact}
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$
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g_3(\Sigma) = \frac{1}{2} b_1 (\Sigma) =
\frac{1}{2} \dim_{\mathbb{R}}H_1(\Sigma, \mathbb{R}),
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$
where $b_1$ is first Betti number of a surface $\Sigma$.
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\end{fact}
\subsection{Seifert matrix}
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
$\alpha_1, \dots, \alpha_n$.
\noindent
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.
\noindent
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]
\fontsize{20}{10}\selectfont
\centering
\def\svgwidth{\linewidth}
\resizebox{0.8\textwidth}{!}{
\input{images/seifert_matrix.pdf_tex}
}
\caption{
A basis $\alpha_1, \alpha_2$
of the first homology
group of a Seifert surface
and a copy of
element $\alpha_1$ pushed up
along vector normal to the Seifert surface.
}
\label{fig:alpha_plus}
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\end{figure}
\begin{theorem}
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:
\begin{enumerate}[label={(\arabic*)}]
\item
$V \rightarrow AVA^T$,
where $A$ is a matrix
with integer coefficients,
\item
$V \rightarrow
\begin{pmatrix}
\begin{array}{c|c}
V &
\begin{matrix}
\ast & 0 \\
\sdots & \sdots\\
\ast & 0
\end{matrix} \\
\hline
\begin{matrix}
\ast & \dots & \ast\\
0 & \dots & 0
\end{matrix}
&
\begin{matrix}
0 & 0\\
1 & 0
\end{matrix}
\end{array}
\end{pmatrix} \quad$
or
$\quad
V \rightarrow
\begin{pmatrix}
\begin{array}{c|c}
V &
\begin{matrix}
\ast & 0 \\
\sdots & \sdots\\
\ast & 0
\end{matrix} \\
\hline
\begin{matrix}
\ast & \dots & \ast\\
0 & \dots & 0
\end{matrix}
&
\begin{matrix}
0 & 1\\
0 & 0
\end{matrix}
\end{array}
\end{pmatrix},$
\item
inverse of (2).
\end{enumerate}
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\end{theorem}