corrections of the first lecture after first review

This commit is contained in:
Maria Marchwicka 2019-10-31 09:20:02 +01:00
parent f257b26d0a
commit 60d7f7d850
16 changed files with 1572 additions and 242 deletions

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@ -2,7 +2,7 @@
%\begin{theorem}
%For any knot $K \subset S^3$ there exists a connected, compact and orientable surface $\Sigma(K)$ such that $\partial \Sigma(K) = K$
%\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:
\begin{align*}
H^1(S^3 \setminus N ) \cong H^1(S^3 \setminus K).

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@ -1,69 +1,104 @@
\begin{definition}
A knot $K$ in $S^3$ is a smooth (PL - smooth) embedding of a circle $S^1$ in $S^3$:
\begin{align*}
\varphi: S^1 \hookrightarrow S^3
\end{align*}
A knot $K$ in $S^3$ is a smooth (PL - smooth)
embedding of a circle $S^1$ in $S^3$:
\[
\varphi: S^1 \hookrightarrow S^3
\]
\end{definition}
\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}.
\begin{example}
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}.
\begin{figure}[h]
\includegraphics[width=0.08\textwidth]
{unknot.png}
\caption{Knots examples: unknot (left) and trefoil (right).}
\label{fig:unknot}
\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}
\end{figure}
\begin{figure}[h]
\includegraphics[width=0.08\textwidth]
{unknot.png}
\caption{Knots examples: unknot (left) and trefoil (right).}
\label{fig:notknot}
\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}
\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}
%\hfill\\
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
$\Phi_1 = \varphi_1$.
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
$\Phi_1 = \varphi_1$.
\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*}
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*}
\end{theorem}
\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}
\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}
\begin{example}
@ -85,20 +120,31 @@ a Borromean link:
%
%
%
\begin{definition}
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: \includegraphics[width=0.05\textwidth]{LinkDiagram1.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}
\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}
\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
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.
\end{fact}
\end{lemma}
\noindent
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}
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}.
\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).
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}
%
%
@ -134,125 +182,201 @@ deformed into each other by a finite sequence of Reidemeister moves (and isotopy
\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.
\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 $\partial \Sigma = L$.\\
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]
\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}
}
\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}
}
\end{figure}
\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 $\partial D_2$.
\begin{figure}[h]
\begin{center}
\includegraphics[width=0.6\textwidth]{seifert_connect.png}
\end{center}
\caption{Connecting two surfaces.}
\label{fig:SeifertConnect}
\centering
\includegraphics[width=0.6\textwidth]
{seifert_connect.png}
\caption{Connecting two surfaces.}
\label{fig:SeifertConnect}
\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.
\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.
\end{theorem}
%
\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}
}
\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}
\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 $K$.
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}
\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}
\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}
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: $\Lk(\alpha, \beta) = \frac{1}{2}(N_+ - N_-)$.
\end{definition}
\begin{definition}
\label{def:lk_via_homo}
Let $\alpha$ and $\beta$ be two disjoint simple cross 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 $\beta$:
\[
\alpha \in H_1(S^3 \setminus \nu(\beta), \mathbb{Z}) \cong \mathbb{Z}.\]
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:
$\Lk(\alpha, \beta) =
\frac{1}{2}(N_+ - N_-)$.
\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}
\begin{example}
\begin{itemize}
\item
A Hopf link:
\begin{figure}[h]
\fontsize{20}{10}\selectfont
\centering{
\def\svgwidth{\linewidth}
\resizebox{0.4\textwidth}{!}{\input{images/linking_hopf.pdf_tex}},
}
\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}
\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}
$
g_3(\Sigma) = \frac{1}{2} b_1 (\Sigma) =
\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}
\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 $\alpha_1^+, \dots \alpha_n^+$ 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.
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.
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.
\begin{figure}[h]
\fontsize{20}{10}\selectfont
\centering{
\def\svgwidth{\linewidth}
\resizebox{0.8\textwidth}{!}{\input{images/seifert_matrix.pdf_tex}}
}
\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}
\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}
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}
@ -271,11 +395,11 @@ $V \rightarrow
1 & 0
\end{matrix}
\end{array}
\end{pmatrix} \quad$
or
$\quad
V \rightarrow
\begin{pmatrix}
\end{pmatrix} \quad$
or
$\quad
V \rightarrow
\begin{pmatrix}
\begin{array}{c|c}
V &
\begin{matrix}
@ -294,9 +418,8 @@ V \rightarrow
0 & 0
\end{matrix}
\end{array}
\end{pmatrix},$
\item
inverse of (2).
\end{enumerate}
\end{pmatrix},$
\item
inverse of (2).
\end{enumerate}
\end{theorem}

48
lec_mess.tex Normal file
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@ -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}

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@ -9,7 +9,9 @@
\usepackage[english]{babel}
\usepackage{caption}
\usepackage[margin=1 cm]{caption}
\usepackage{subcaption}
%\usepackage{cleveref} - after hyperref
\usepackage{comment}
\usepackage{csquotes}
@ -21,6 +23,7 @@
\usepackage{graphicx}
\usepackage{hyperref}
\usepackage[nameinlink]{cleveref}
\usepackage{mathtools}
@ -28,6 +31,7 @@
\usepackage[section]{placeins}
\usepackage[pdf]{pstricks}
%\usepackage{subcaption} % added after caption
\usepackage{tikz}
\usepackage{titlesec}
@ -127,7 +131,7 @@
\begin{document}
\tableofcontents
%\newpage
\newpage
%\input{myNotes}
\section{Basic definitions
@ -214,7 +218,7 @@ Surgery \hfill\DTMdate{2019-06-03}}
\texorpdfstring{
\hfill\DTMdate{2019-06-17}}
{}}
\input{mess.tex}
\input{lec_mess.tex}
\end{document}