corrections of the first lecture after first review
This commit is contained in:
Maria Marchwicka
parent
f257b26d0a
commit
60d7f7d850
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|||||||
%\begin{theorem}
|
%\begin{theorem}
|
||||||
%For any knot $K \subset S^3$ there exists a connected, compact and orientable surface $\Sigma(K)$ such that $\partial \Sigma(K) = K$
|
%For any knot $K \subset S^3$ there exists a connected, compact and orientable surface $\Sigma(K)$ such that $\partial \Sigma(K) = K$
|
||||||
%\end{theorem}
|
%\end{theorem}
|
||||||
\begin{proof}(Theorem \ref{theo:Seifert})\\
|
\begin{proof}(\Cref{theo:Seifert})\\
|
||||||
Let $K \in S^3$ be a knot and $N = \nu(K)$ be its tubular neighbourhood. Because $K$ and $N$ are homotopy equivalent, we get:
|
Let $K \in S^3$ be a knot and $N = \nu(K)$ be its tubular neighbourhood. Because $K$ and $N$ are homotopy equivalent, we get:
|
||||||
\begin{align*}
|
\begin{align*}
|
||||||
H^1(S^3 \setminus N ) \cong H^1(S^3 \setminus K).
|
H^1(S^3 \setminus N ) \cong H^1(S^3 \setminus K).
|
||||||
|
|||||||
495
lec_25_02.tex
495
lec_25_02.tex
@ -1,69 +1,104 @@
|
|||||||
\begin{definition}
|
\begin{definition}
|
||||||
A knot $K$ in $S^3$ is a smooth (PL - smooth) embedding of a circle $S^1$ in $S^3$:
|
A knot $K$ in $S^3$ is a smooth (PL - smooth)
|
||||||
\begin{align*}
|
embedding of a circle $S^1$ in $S^3$:
|
||||||
\varphi: S^1 \hookrightarrow S^3
|
\[
|
||||||
\end{align*}
|
\varphi: S^1 \hookrightarrow S^3
|
||||||
|
\]
|
||||||
\end{definition}
|
\end{definition}
|
||||||
|
|
||||||
\noindent
|
\noindent
|
||||||
Usually we think about a knot as an image of an embedding: $K = \varphi(S^1)$.
|
Usually we think about a knot
|
||||||
Some basic examples and counterexamples are shown respectively in \autoref{fig:unknot} and \autoref{fig:notknot}.
|
as an image of an embedding:
|
||||||
\begin{example}
|
$K = \varphi(S^1)$.
|
||||||
|
Some basic examples and counterexamples
|
||||||
|
are shown respectively in
|
||||||
|
\autoref{fig:unknot} and
|
||||||
|
\autoref{fig:notknot}.
|
||||||
|
|
||||||
\begin{figure}[h]
|
\begin{figure}[h]
|
||||||
\includegraphics[width=0.08\textwidth]
|
\centering
|
||||||
{unknot.png}
|
\begin{subfigure}{0.3\textwidth}
|
||||||
\caption{Knots examples: unknot (left) and trefoil (right).}
|
\centering
|
||||||
\label{fig:unknot}
|
\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}
|
||||||
\end{figure}
|
\end{figure}
|
||||||
|
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\begin{figure}[h]
|
\begin{figure}[h]
|
||||||
\includegraphics[width=0.08\textwidth]
|
\centering
|
||||||
{unknot.png}
|
\begin{subfigure}{0.3\textwidth}
|
||||||
\caption{Knots examples: unknot (left) and trefoil (right).}
|
\centering
|
||||||
\label{fig:notknot}
|
\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}
|
||||||
\end{figure}
|
\end{figure}
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
\begin{itemize}
|
|
||||||
\item
|
|
||||||
Knots:
|
|
||||||
\includegraphics[width=0.08\textwidth]{unknot.png} (unknot),
|
|
||||||
\includegraphics[width=0.08\textwidth]{trefoil.png} (trefoil).
|
|
||||||
\item
|
|
||||||
Not knots:
|
|
||||||
\includegraphics[width=0.12\textwidth]{not_injective_knot.png}
|
|
||||||
(it is not an injection),
|
|
||||||
\includegraphics[width=0.08\textwidth]{not_smooth_knot.png}
|
|
||||||
(it is not smooth).
|
|
||||||
\end{itemize}
|
|
||||||
\end{example}
|
|
||||||
\begin{definition}
|
\begin{definition}
|
||||||
%\hfill\\
|
Two knots $K_0 = \varphi_0(S^1)$,
|
||||||
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
|
$K_1 = \varphi_1(S^1)$
|
||||||
\begin{align*}
|
are equivalent if the embeddings
|
||||||
&\Phi: S^1 \times [0, 1] \hookrightarrow S^3, \\
|
$\varphi_0$ and $\varphi_1$ are isotopic,
|
||||||
&\Phi(x, t) = \Phi_t(x)
|
that is there exists a continues function
|
||||||
\end{align*}
|
\begin{align*}
|
||||||
such that $\Phi_t$ is an embedding for any $t \in [0,1]$, $\Phi_0 = \varphi_0$ and
|
&\Phi: S^1 \times
|
||||||
$\Phi_1 = \varphi_1$.
|
[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
|
||||||
|
$\Phi_1 = \varphi_1$.
|
||||||
\end{definition}
|
\end{definition}
|
||||||
|
|
||||||
\begin{theorem}
|
\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:
|
Two knots $K_0$ and $K_1$ are isotopic
|
||||||
\begin{align*}
|
if and only if they are ambient isotopic,
|
||||||
&\psi(t) = \psi_t \text{ is continius on $t\in [0,1]$},\\
|
i.e. there exists a family of self-diffeomorphisms
|
||||||
&\psi_t: S^3 \hookrightarrow S^3,\\
|
$\Psi = \{\psi_t: t \in [0, 1]\}$ such that:
|
||||||
& \psi_0 = id ,\\
|
\begin{align*}
|
||||||
& \psi_1(K_0) = K_1.
|
&\psi(t) = \psi_t
|
||||||
\end{align*}
|
\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*}
|
||||||
\end{theorem}
|
\end{theorem}
|
||||||
|
|
||||||
\begin{definition}
|
\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 $S^1$.
|
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 $S^1$.
|
||||||
\end{definition}
|
\end{definition}
|
||||||
|
|
||||||
\begin{definition}
|
\begin{definition}
|
||||||
A link with k - components is a (smooth) embedding of $\overbrace{S^1 \sqcup \ldots \sqcup S^1}^k$ in $S^3$.
|
A link with $k$ - components is a
|
||||||
|
(smooth) embedding of
|
||||||
|
$\overbrace{S^1 \sqcup \ldots \sqcup S^1}^k$
|
||||||
|
in $S^3$.
|
||||||
\end{definition}
|
\end{definition}
|
||||||
|
|
||||||
\begin{example}
|
\begin{example}
|
||||||
@ -85,20 +120,31 @@ a Borromean link:
|
|||||||
%
|
%
|
||||||
%
|
%
|
||||||
%
|
%
|
||||||
\begin{definition}
|
\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:
|
A link diagram $D_{\pi}$ is a picture
|
||||||
\begin{enumerate}[label={(\arabic*)}]
|
over projection $\pi$ of a link $L$ in
|
||||||
\item
|
$\mathbb{R}^3$($S^3$) to
|
||||||
$D_{\pi |_L}$ is non degenerate: \includegraphics[width=0.05\textwidth]{LinkDiagram1.png},
|
$\mathbb{R}^2$ ($S^2$) such that:
|
||||||
\item the double points are not degenerate: \includegraphics[width=0.03\textwidth]{LinkDiagram2.png},
|
\begin{enumerate}[label={(\arabic*)}]
|
||||||
\item there are no triple point: \includegraphics[width=0.05\textwidth]{LinkDiagram3.png}.
|
\item
|
||||||
\end{enumerate}
|
$D_{\pi |_L}$ is non degenerate,
|
||||||
|
\item
|
||||||
|
the double points are not degenerate,
|
||||||
|
\item there are no triple point.
|
||||||
|
\end{enumerate}
|
||||||
\end{definition}
|
\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}.
|
||||||
|
|
||||||
|
|
||||||
\noindent
|
\noindent
|
||||||
There are under- and overcrossings (tunnels and bridges) on a link diagrams with an obvious meaning.
|
There are under- and overcrossings (tunnels and bridges) on a link diagrams with an obvious meaning.
|
||||||
\begin{fact}
|
\begin{lemma}
|
||||||
Every link admits a link diagram.
|
Every link admits a link diagram.
|
||||||
\end{fact}
|
\end{lemma}
|
||||||
\noindent
|
\noindent
|
||||||
|
|
||||||
Let $D$ be a diagram of an oriented link (to each component of a link we add an arrow in the diagram).
|
Let $D$ be a diagram of an oriented link (to each component of a link we add an arrow in the diagram).
|
||||||
@ -108,15 +154,17 @@ $\left(\PICorientpluscross\right)$, called a positive crossing, and left-handed
|
|||||||
\subsection{Reidemeister moves}
|
\subsection{Reidemeister moves}
|
||||||
A Reidemeister move is one of the three types of operation on a link diagram as shown below:
|
A Reidemeister move is one of the three types of operation on a link diagram as shown below:
|
||||||
\begin{enumerate}[label=\Roman*]
|
\begin{enumerate}[label=\Roman*]
|
||||||
\item\hfill\\
|
\item\hfill\\
|
||||||
\includegraphics[width=0.6\textwidth]{rm1.png},
|
\includegraphics[width=0.6\textwidth]{rm1.png},
|
||||||
\item\hfill\\\includegraphics[width=0.6\textwidth]{rm2.png},
|
\item\hfill\\\includegraphics[width=0.6\textwidth]{rm2.png},
|
||||||
\item\hfill\\\includegraphics[width=0.4\textwidth]{rm3.png}.
|
\item\hfill\\\includegraphics[width=0.4\textwidth]{rm3.png}.
|
||||||
\end{enumerate}
|
\end{enumerate}
|
||||||
|
|
||||||
\begin{theorem} [Reidemeister, 1927 ]
|
\begin{theorem} [Reidemeister, 1927 ]
|
||||||
Two diagrams of the same link can be
|
Two diagrams of the same link can
|
||||||
deformed into each other by a finite sequence of Reidemeister moves (and isotopy of the plane).
|
be deformed into each other by a finite
|
||||||
|
sequence of Reidemeister moves
|
||||||
|
(and isotopy of the plane).
|
||||||
\end{theorem}
|
\end{theorem}
|
||||||
%
|
%
|
||||||
%
|
%
|
||||||
@ -134,169 +182,244 @@ deformed into each other by a finite sequence of Reidemeister moves (and isotopy
|
|||||||
\noindent
|
\noindent
|
||||||
Let $D$ be an oriented diagram of a link $L$. We change the diagram by smoothing each crossing:
|
Let $D$ be an oriented diagram of a link $L$. We change the diagram by smoothing each crossing:
|
||||||
\begin{align*}
|
\begin{align*}
|
||||||
\PICorientpluscross \mapsto \PICorientLRsplit,\\
|
\PICorientpluscross \mapsto
|
||||||
\PICorientminuscross \mapsto \PICorientLRsplit.
|
\PICorientLRsplit,\\
|
||||||
|
\PICorientminuscross \mapsto
|
||||||
|
\PICorientLRsplit.
|
||||||
\end{align*}
|
\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 $\partial \Sigma = L$.\\
|
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 $\partial \Sigma = L$.\\
|
||||||
|
|
||||||
\begin{figure}[h]
|
\begin{figure}[h]
|
||||||
\fontsize{15}{10}\selectfont
|
\fontsize{15}{10}\selectfont
|
||||||
\centering{
|
\centering{
|
||||||
\def\svgwidth{\linewidth}
|
\def\svgwidth{\linewidth}
|
||||||
\resizebox{0.8\textwidth}{!}{\input{images/seifert_alg.pdf_tex}}
|
\resizebox{0.8\textwidth}{!}
|
||||||
\caption{Constructing a Seifert surface.}
|
{\input{images/seifert_alg.pdf_tex}}
|
||||||
\label{fig:SeifertAlg}
|
\caption{Constructing a Seifert surface.}
|
||||||
}
|
\label{fig:SeifertAlg}
|
||||||
|
}
|
||||||
\end{figure}
|
\end{figure}
|
||||||
|
|
||||||
\noindent
|
\noindent
|
||||||
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 $\partial D_2$.
|
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 $\partial D_2$.
|
||||||
|
|
||||||
\begin{figure}[h]
|
\begin{figure}[h]
|
||||||
\begin{center}
|
\centering
|
||||||
\includegraphics[width=0.6\textwidth]{seifert_connect.png}
|
\includegraphics[width=0.6\textwidth]
|
||||||
\end{center}
|
{seifert_connect.png}
|
||||||
\caption{Connecting two surfaces.}
|
\caption{Connecting two surfaces.}
|
||||||
\label{fig:SeifertConnect}
|
\label{fig:SeifertConnect}
|
||||||
\end{figure}
|
\end{figure}
|
||||||
|
|
||||||
\begin{theorem}[Seifert]
|
\begin{theorem}[Seifert]\label{theo:Seifert}
|
||||||
\label{theo:Seifert}
|
Every link in $S^3$ bounds a surface
|
||||||
Every link in $S^3$ bounds a surface $\Sigma$ that is compact, connected and orientable. Such a surface is called a Seifert surface.
|
$\Sigma$ that is compact, connected
|
||||||
|
and orientable.
|
||||||
|
Such a surface is called a Seifert surface.
|
||||||
\end{theorem}
|
\end{theorem}
|
||||||
%
|
|
||||||
\begin{figure}[h]
|
\begin{figure}[h]
|
||||||
\fontsize{12}{10}\selectfont
|
\fontsize{12}{10}\selectfont
|
||||||
\centering{
|
\centering
|
||||||
\def\svgwidth{\linewidth}
|
\def\svgwidth{\linewidth}
|
||||||
\resizebox{1\textwidth}{!}{\input{images/torus_1_2_3.pdf_tex}}
|
\resizebox{1\textwidth}{!}{
|
||||||
\caption{Genus of an orientable surface.}
|
\input{images/torus_1_2_3.pdf_tex}}
|
||||||
\label{fig:genera}
|
\caption{Genus of an orientable surface.}
|
||||||
}
|
\label{fig:genera}
|
||||||
\end{figure}
|
\end{figure}
|
||||||
%
|
%
|
||||||
%
|
%
|
||||||
\begin{definition}
|
\begin{definition}
|
||||||
The three genus $g_3(K)$ ($g(K)$) of a knot $K$ is the minimal genus of a Seifert surface $\Sigma$ for $K$.
|
The three genus $g_3(K)$ ($g(K)$)
|
||||||
|
of a knot $K$ is the minimal genus
|
||||||
|
of a Seifert surface $\Sigma$ for $K$.
|
||||||
\end{definition}
|
\end{definition}
|
||||||
|
|
||||||
\begin{corollary}
|
\begin{corollary}
|
||||||
A knot $K$ is trivial if and only $g_3(K) = 0$.
|
A knot $K$ is trivial if and only
|
||||||
|
$g_3(K) = 0$.
|
||||||
\end{corollary}
|
\end{corollary}
|
||||||
|
|
||||||
\noindent
|
\noindent
|
||||||
Remark: there are knots that admit non isotopic Seifert surfaces of minimal genus (András Juhász, 2008).
|
Remark: there are knots that admit non isotopic
|
||||||
|
Seifert surfaces of minimal genus
|
||||||
|
(András Juhász, 2008).
|
||||||
|
|
||||||
\begin{definition}
|
\begin{definition}
|
||||||
Suppose $\alpha$ and $\beta$ are two simple closed curves in $\mathbb{R}^3$.
|
Suppose $\alpha$ and $\beta$ are two
|
||||||
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: $\Lk(\alpha, \beta) = \frac{1}{2}(N_+ - N_-)$.
|
simple closed curves in $\mathbb{R}^3$.
|
||||||
\end{definition}
|
On a diagram $L$ consider all crossings
|
||||||
\begin{definition}
|
between $\alpha$ and $\beta$.
|
||||||
\label{def:lk_via_homo}
|
Let $N_+$ be the number
|
||||||
Let $\alpha$ and $\beta$ be two disjoint simple cross curves in $S^3$.
|
of positive crossings,
|
||||||
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 $\beta$:
|
$N_-$ - negative.
|
||||||
\[
|
Then the linking number:
|
||||||
\alpha \in H_1(S^3 \setminus \nu(\beta), \mathbb{Z}) \cong \mathbb{Z}.\]
|
$\Lk(\alpha, \beta) =
|
||||||
|
\frac{1}{2}(N_+ - N_-)$.
|
||||||
|
\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 $\beta$:
|
||||||
|
\[
|
||||||
|
\alpha \in H_1(S^3 \setminus
|
||||||
|
\nu(\beta), \mathbb{Z})
|
||||||
|
\cong \mathbb{Z}.
|
||||||
|
\]
|
||||||
\end{definition}
|
\end{definition}
|
||||||
|
|
||||||
\begin{example}
|
|
||||||
\begin{itemize}
|
|
||||||
\item
|
|
||||||
A Hopf link:
|
|
||||||
\begin{figure}[h]
|
\begin{figure}[h]
|
||||||
\fontsize{20}{10}\selectfont
|
\fontsize{10}{8}\selectfont
|
||||||
\centering{
|
\centering
|
||||||
\def\svgwidth{\linewidth}
|
\def\svgwidth{\linewidth}
|
||||||
\resizebox{0.4\textwidth}{!}{\input{images/linking_hopf.pdf_tex}},
|
\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}
|
||||||
\end{figure}
|
\end{figure}
|
||||||
\item
|
|
||||||
$T(6, 2)$ link:
|
|
||||||
\begin{figure}[h]
|
|
||||||
\fontsize{20}{10}\selectfont
|
|
||||||
\centering{
|
|
||||||
\def\svgwidth{\linewidth}
|
|
||||||
\resizebox{0.4\textwidth}{!}{\input{images/linking_torus_6_2.pdf_tex}}.
|
|
||||||
}
|
|
||||||
\end{figure}
|
|
||||||
\end{itemize}
|
|
||||||
\end{example}
|
|
||||||
\begin{fact}
|
\begin{fact}
|
||||||
$
|
$
|
||||||
g_3(\Sigma) = \frac{1}{2} b_1 (\Sigma) =
|
g_3(\Sigma) = \frac{1}{2} b_1 (\Sigma) =
|
||||||
\frac{1}{2} \dim_{\mathbb{R}}H_1(\Sigma, \mathbb{R}),
|
\frac{1}{2} \dim_{\mathbb{R}}H_1(\Sigma, \mathbb{R}),
|
||||||
$
|
$
|
||||||
where $b_1$ is first Betti number of $\Sigma$.
|
where $b_1$ is first Betti number of a surface $\Sigma$.
|
||||||
\end{fact}
|
\end{fact}
|
||||||
|
|
||||||
\subsection{Seifert matrix}
|
\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 $\alpha_1, \dots, \alpha_n$.
|
Let $L$ be a link and
|
||||||
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.
|
$\Sigma$ be an oriented
|
||||||
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.
|
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.
|
||||||
|
|
||||||
\begin{figure}[h]
|
\begin{figure}[h]
|
||||||
\fontsize{20}{10}\selectfont
|
\fontsize{20}{10}\selectfont
|
||||||
\centering{
|
\centering
|
||||||
\def\svgwidth{\linewidth}
|
\def\svgwidth{\linewidth}
|
||||||
\resizebox{0.8\textwidth}{!}{\input{images/seifert_matrix.pdf_tex}}
|
\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}
|
||||||
\end{figure}
|
\end{figure}
|
||||||
|
|
||||||
\begin{theorem}
|
\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:
|
The Seifert matrices $S_1$ and $S_2$
|
||||||
\begin{enumerate}[label={(\arabic*)}]
|
for the same link $L$ are S-equivalent,
|
||||||
|
that is, $S_2$ can be obtained from
|
||||||
\item
|
$S_1$ by a sequence of following moves:
|
||||||
$V \rightarrow AVA^T$, where $A$ is a matrix with integer coefficients,
|
\begin{enumerate}[label={(\arabic*)}]
|
||||||
|
\item
|
||||||
\item
|
$V \rightarrow AVA^T$,
|
||||||
|
where $A$ is a matrix
|
||||||
$V \rightarrow
|
with integer coefficients,
|
||||||
\begin{pmatrix}
|
\item
|
||||||
\begin{array}{c|c}
|
$V \rightarrow
|
||||||
V &
|
\begin{pmatrix}
|
||||||
\begin{matrix}
|
\begin{array}{c|c}
|
||||||
\ast & 0 \\
|
V &
|
||||||
\sdots & \sdots\\
|
\begin{matrix}
|
||||||
\ast & 0
|
\ast & 0 \\
|
||||||
\end{matrix} \\
|
\sdots & \sdots\\
|
||||||
\hline
|
\ast & 0
|
||||||
\begin{matrix}
|
\end{matrix} \\
|
||||||
\ast & \dots & \ast\\
|
\hline
|
||||||
0 & \dots & 0
|
\begin{matrix}
|
||||||
\end{matrix}
|
\ast & \dots & \ast\\
|
||||||
&
|
0 & \dots & 0
|
||||||
\begin{matrix}
|
\end{matrix}
|
||||||
0 & 0\\
|
&
|
||||||
1 & 0
|
\begin{matrix}
|
||||||
\end{matrix}
|
0 & 0\\
|
||||||
\end{array}
|
1 & 0
|
||||||
\end{pmatrix} \quad$
|
\end{matrix}
|
||||||
or
|
\end{array}
|
||||||
$\quad
|
\end{pmatrix} \quad$
|
||||||
V \rightarrow
|
or
|
||||||
\begin{pmatrix}
|
$\quad
|
||||||
\begin{array}{c|c}
|
V \rightarrow
|
||||||
V &
|
\begin{pmatrix}
|
||||||
\begin{matrix}
|
\begin{array}{c|c}
|
||||||
\ast & 0 \\
|
V &
|
||||||
\sdots & \sdots\\
|
\begin{matrix}
|
||||||
\ast & 0
|
\ast & 0 \\
|
||||||
\end{matrix} \\
|
\sdots & \sdots\\
|
||||||
\hline
|
\ast & 0
|
||||||
\begin{matrix}
|
\end{matrix} \\
|
||||||
\ast & \dots & \ast\\
|
\hline
|
||||||
0 & \dots & 0
|
\begin{matrix}
|
||||||
\end{matrix}
|
\ast & \dots & \ast\\
|
||||||
&
|
0 & \dots & 0
|
||||||
\begin{matrix}
|
\end{matrix}
|
||||||
0 & 1\\
|
&
|
||||||
0 & 0
|
\begin{matrix}
|
||||||
\end{matrix}
|
0 & 1\\
|
||||||
\end{array}
|
0 & 0
|
||||||
\end{pmatrix},$
|
\end{matrix}
|
||||||
\item
|
\end{array}
|
||||||
inverse of (2).
|
\end{pmatrix},$
|
||||||
|
\item
|
||||||
\end{enumerate}
|
inverse of (2).
|
||||||
|
\end{enumerate}
|
||||||
\end{theorem}
|
\end{theorem}
|
||||||
|
|||||||
48
lec_mess.tex
Normal file
48
lec_mess.tex
Normal file
@ -0,0 +1,48 @@
|
|||||||
|
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
|
\begin{fact}[Milnor Singular Points of Complex Hypersurfaces]
|
||||||
|
\end{fact}
|
||||||
|
%\end{comment}
|
||||||
|
\noindent
|
||||||
|
An oriented knot is called negative amphichiral if the mirror image $m(K)$ of $K$ is equivalent the reverse knot of $K$: $K^r$. \\
|
||||||
|
\begin{problem}
|
||||||
|
Prove that if $K$ is negative amphichiral, then $K \# K = 0$ in
|
||||||
|
$\mathscr{C}$.
|
||||||
|
%
|
||||||
|
%\\
|
||||||
|
%Hint: $ -K = m(K)^r = (K^r)^r = K$
|
||||||
|
\end{problem}
|
||||||
|
\begin{example}
|
||||||
|
Figure 8 knot is negative amphichiral.
|
||||||
|
\end{example}
|
||||||
|
%
|
||||||
|
%
|
||||||
|
\begin{theorem}
|
||||||
|
Let $H_p$ be a $p$ - torsion part of $H$. There exists an orthogonal decomposition of $H_p$:
|
||||||
|
\[
|
||||||
|
H_p = H_{p, 1} \oplus \dots \oplus H_{p, r_p}.
|
||||||
|
\]
|
||||||
|
$H_{p, i}$ is a cyclic module:
|
||||||
|
\[
|
||||||
|
H_{p, i} = \quot{\mathbb{Z}[t, t^{-1}]}{p^{k_i} \mathbb{Z} [t, t^{-1}]}
|
||||||
|
\]
|
||||||
|
\end{theorem}
|
||||||
|
\noindent
|
||||||
|
The proof is the same as over $\mathbb{Z}$.
|
||||||
|
\noindent
|
||||||
|
%Add NotePrintSaveCiteYour opinionEmailShare
|
||||||
|
%Saveliev, Nikolai
|
||||||
|
|
||||||
|
%Lectures on the Topology of 3-Manifolds
|
||||||
|
%An Introduction to the Casson Invariant
|
||||||
|
|
||||||
|
\begin{figure}[h]
|
||||||
|
\fontsize{10}{10}\selectfont
|
||||||
|
\centering{
|
||||||
|
\def\svgwidth{\linewidth}
|
||||||
|
\resizebox{0.5\textwidth}{!}{\input{images/ball_4_alpha_beta.pdf_tex}}
|
||||||
|
}
|
||||||
|
%\caption{Sketch for Fact %%\label{fig:concordance_m}
|
||||||
|
\end{figure}
|
||||||
Binary file not shown.
@ -9,7 +9,9 @@
|
|||||||
|
|
||||||
\usepackage[english]{babel}
|
\usepackage[english]{babel}
|
||||||
|
|
||||||
\usepackage{caption}
|
\usepackage[margin=1 cm]{caption}
|
||||||
|
\usepackage{subcaption}
|
||||||
|
%\usepackage{cleveref} - after hyperref
|
||||||
\usepackage{comment}
|
\usepackage{comment}
|
||||||
\usepackage{csquotes}
|
\usepackage{csquotes}
|
||||||
|
|
||||||
@ -21,6 +23,7 @@
|
|||||||
|
|
||||||
\usepackage{graphicx}
|
\usepackage{graphicx}
|
||||||
\usepackage{hyperref}
|
\usepackage{hyperref}
|
||||||
|
\usepackage[nameinlink]{cleveref}
|
||||||
|
|
||||||
\usepackage{mathtools}
|
\usepackage{mathtools}
|
||||||
|
|
||||||
@ -28,6 +31,7 @@
|
|||||||
\usepackage[section]{placeins}
|
\usepackage[section]{placeins}
|
||||||
\usepackage[pdf]{pstricks}
|
\usepackage[pdf]{pstricks}
|
||||||
|
|
||||||
|
%\usepackage{subcaption} % added after caption
|
||||||
\usepackage{tikz}
|
\usepackage{tikz}
|
||||||
\usepackage{titlesec}
|
\usepackage{titlesec}
|
||||||
|
|
||||||
@ -127,7 +131,7 @@
|
|||||||
|
|
||||||
\begin{document}
|
\begin{document}
|
||||||
\tableofcontents
|
\tableofcontents
|
||||||
%\newpage
|
\newpage
|
||||||
%\input{myNotes}
|
%\input{myNotes}
|
||||||
|
|
||||||
\section{Basic definitions
|
\section{Basic definitions
|
||||||
@ -214,7 +218,7 @@ Surgery \hfill\DTMdate{2019-06-03}}
|
|||||||
\texorpdfstring{
|
\texorpdfstring{
|
||||||
\hfill\DTMdate{2019-06-17}}
|
\hfill\DTMdate{2019-06-17}}
|
||||||
{}}
|
{}}
|
||||||
\input{mess.tex}
|
\input{lec_mess.tex}
|
||||||
|
|
||||||
\end{document}
|
\end{document}
|
||||||
|
|
||||||
|
|||||||
Loading…
Reference in New Issue
Block a user