correction of first lecture starts

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Maria Marchwicka 2019-10-25 07:02:17 +02:00
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@ -69,5 +69,31 @@ S^1 \times \pt &\longrightarrow \pt \times S^1 \\
} }
\end{figure} \end{figure}
\begin{proof}[Sketch of proof]
We will show that each diffeomorphism is isotopic to $\begin{pmatrix}
p & q\\
r & s
\end{pmatrix}$.
\begin{equation*}
\quot{\Diff_+(S^1 \times S^1)}{\Iso(S^1 \times S^1)} = \mcg(S^1 \times S^1) = \Sl(2, \mathbb{Z})
\end{equation*}
\begin{figure}[h]
\fontsize{20}{10}\selectfont
\centering{
\def\svgwidth{\linewidth}
\resizebox{0.4\textwidth}{!}{\input{images/torus_mu_lambda.pdf_tex}}
\caption{Choice of meridian and longitude.}
\label{fig:torus_twist}
}
\end{figure}
\end{proof}
Let $N = D^2 \times S$ be a tubular neighbourhood of a knot $K$. Consider its boundary $\partial N = S^1 \times S^1$. There exists a simple closed curve $\mu \subset \partial N$ (a meridian) that bounds a disk in $N$. We choose another simple closed curve $\lambda$ (a longitude) so that $\Lk(\lambda, K) = 0$. \\
????????
\\
$\lambda \mu = 1 $ intersection\\
$\pi_0 (\Gl(2, \mathbb{R})$\\
???????????
\\
In other words a homotopy class: $[\lambda] = 0$ in $H_1(S^3 \setminus N, \mathbb{Z})$.

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@ -1,4 +1,4 @@
\subsection{Existence of Seifert surface - second proof} \subsection{Existence of a Seifert surface - second proof}
%\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}
@ -209,7 +209,8 @@ There are not trivial knots with Alexander polynomial equal $1$, for example:
$\Delta_{11n34} \equiv 1$. $\Delta_{11n34} \equiv 1$.
\end{example} \end{example}
\subsection{Decomposition of $3$-sphere} \subsection{Decomposition of \texorpdfstring{
$3$-sphere}{3-sphere}}
We know that $3$ - sphere can be obtained by gluing two solid tori: We know that $3$ - sphere can be obtained by gluing two solid tori:
\[ \[
S^3 = \partial D^4 = \partial (D^2 \times D^2) = (D^2 \times S^1) \cup (S^1 \times D^2). S^3 = \partial D^4 = \partial (D^2 \times D^2) = (D^2 \times S^1) \cup (S^1 \times D^2).

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@ -17,12 +17,17 @@ The infinite cyclic cover of a knot complement $X$ is the cover associated with
\label{fig:covering} \label{fig:covering}
} }
\end{figure} \end{figure}
\noindent
\subsection{Double branched cover.}
Let $K \subset S^3$ be a knot and $\Sigma$
its Seifert surface.
Let us consider a knot complement $S^3 \setminus N(K)$.
\begin{figure}[h] \begin{figure}[h]
\fontsize{10}{10}\selectfont \fontsize{10}{10}\selectfont
\centering{ \centering{
\def\svgwidth{\linewidth} \def\svgwidth{\linewidth}
\resizebox{0.8\textwidth}{!}{\input{images/knot_complement.pdf_tex}} \resizebox{0.8\textwidth}{!}{\input{images/knot_complement.pdf_tex}}
\caption{A knot complement.} \caption{The double cover of the $3$-sphere branched over a knot $K$.}
\label{fig:complement} \label{fig:complement}
} }
\end{figure} \end{figure}

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@ -1,9 +1,9 @@
$X$ is a closed orientable four-manifold. Assume $\pi_1(X) = 0$ (it is not needed to define the intersection form). In particular $H_1(X) = 0$. $X$ is a closed orientable four-manifold. For simplicity assume $\pi_1(X) = 0$ (it is not needed to define the intersection form). In particular $H_1(X) = 0$.
$H_2$ is free (exercise). $H_2$ is free (exercise).
\begin{align*} \[
H_2(X, \mathbb{Z}) \xrightarrow{\text{Poincar\'e duality}} H^2(X, \mathbb{Z} ) \xrightarrow{\text{evaluation}}\Hom(H_2(X, \mathbb{Z}), \mathbb{Z}) H_2(X, \mathbb{Z}) \xrightarrow{\text{Poincar\'e duality}} H^2(X, \mathbb{Z} ) \xrightarrow{\text{evaluation}}\Hom(H_2(X, \mathbb{Z}), \mathbb{Z}).
\end{align*} \]
\noindent \noindent
Intersection form: Intersection form:
$H_2(X, \mathbb{Z}) \times $H_2(X, \mathbb{Z}) \times
@ -18,7 +18,7 @@ Let $A$ and $B$ be closed, oriented surfaces in $X$.
\resizebox{0.5\textwidth}{!}{\input{images/intersection_form_A_B.pdf_tex}} \resizebox{0.5\textwidth}{!}{\input{images/intersection_form_A_B.pdf_tex}}
} }
\caption{$T_X A + T_X B = T_X X$ \caption{$T_X A + T_X B = T_X X$
}\label{fig:torus_alpha_beta} }\label{fig:intersection}
\end{figure} \end{figure}
??????????????????????? ???????????????????????
\begin{align*} \begin{align*}

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@ -1,48 +1 @@
Consider a surgery
\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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@ -60,7 +60,7 @@ H_1(\widetilde{X}, \mathbb{Z}[t, t^{-1}]) \longrightarrow \quot{\mathbb{Q}}{\mat
\begin{align*} \begin{align*}
H_1(\widetilde{X}, \mathbb{Z}[t, t^{-1}]) \times H_1(\widetilde{X}, \mathbb{Z}[t, t^{-1}]) \times
H_1(\widetilde{X}, \mathbb{Z}[t, t^{-1}]) &\longrightarrow \quot{\mathbb{Q}}{\mathbb{Z}[t, t^{-1}]}\\ H_1(\widetilde{X}, \mathbb{Z}[t, t^{-1}]) &\longrightarrow \quot{\mathbb{Q}}{\mathbb{Z}[t, t^{-1}]}\\
(\alpha, \beta) \quad &\mapsto \alpha^{-1}(t -1)(tV - V^T)^{-1}\beta (\alpha, \beta) \quad &\mapsto \alpha^{-1}{(t -1)(tV - V^T)}^{-1}\beta
\end{align*} \end{align*}
\end{fact} \end{fact}
\noindent \noindent

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@ -6,8 +6,24 @@ A knot $K$ in $S^3$ is a smooth (PL - smooth) embedding of a circle $S^1$ in $S^
\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 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} \begin{example}
\begin{figure}[h]
\includegraphics[width=0.08\textwidth]
{unknot.png}
\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}
\end{figure}
\begin{itemize} \begin{itemize}
\item \item
Knots: Knots:
@ -41,12 +57,15 @@ Two knots $K_0$ and $K_1$ are isotopic if and only if they are ambient isotopic,
& \psi_1(K_0) = K_1. & \psi_1(K_0) = K_1.
\end{align*} \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}
Links: Links:
\begin{itemize} \begin{itemize}

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@ -62,7 +62,80 @@ H_1(\Sigma(K), \mathbb{Z})
(a, b) \mapsto a{(V + V^T)}^{-1} b (a, b) \mapsto a{(V + V^T)}^{-1} b
\end{eqnarray*} \end{eqnarray*}
???????????????????\\ ???????????????????\\
???????????????????\\
\begin{eqnarray*} \begin{eqnarray*}
y \mapsto y + Az \\ y \mapsto y + Az \\
\overline{x^T} A^{-1}(y + Az) = \overline{x^T} A^{-1} + \overline{x^T} \mathbb{1} z \overline{x^T} A^{-1}(y + Az) = \overline{x^T} A^{-1} y + \overline{x^T} \mathbb{1} z = \overline{x^T} A^{-1}y \in \quot{\Omega}{\Lambda} \\
\overline{x^T} \mathbb{1} z \in \Lambda \\
H_1(\widetilde{X}, \Lambda) =
\quot{ \Lambda^n }{(Vt - V) \Lambda^n}
\\
(a, b) \mapsto \overline{a^T}(Vt - V^T)^{-1} (t -1)b
\end{eqnarray*} \end{eqnarray*}
(Blanchfield '59)
\begin{theorem}[Kearton '75, Friedl, Powell '15]
There exits a matrix $A$ representing the Blanchfield paring over $\mathbb{Z}[t, t^{-1}]$. The size of $A$ is a size of Seifert form.
\end{theorem}
Remark:
\begin{enumerate}
\item
Over $\mathbb{R}$ we can take $A$ to be diagonal.
\item
The jump of signature function at $\xi$ is
equal to
\[
\lim_{t \rightarrow 0^+} \sign A (e^{it} \xi) - \sign A(e^{-it} \xi).
\]
\item
The minimal size of a matrix $A$ that presents a Blanchfield paring (over $\mathbb{Z}[t, t^{-1}]$) for a knot $K$ is a knot invariant.
\end{enumerate}
\subsection{The unknotting number}
Let $K$ be a knot and $D$ a knot diagram. A crossing change is a modification of a knot diagram by one of following changes
\begin{align*}
\PICorientpluscross \mapsto \PICorientminuscross ,\\
\PICorientminuscross \mapsto\PICorientminuscross.
\end{align*}
The unknotting number $u(K)$ is a number of crossing changes needed to turn a knot into an unknot, where the minimum is taken over all diagrams of a given knot.
\begin{definition}
A Gordian distance $G(K, K^\prime)$ between knots $K$ and $K^\prime$ is the minimal number of crossing changes required to turn $K$ into $K^\prime$.
\end{definition}
\begin{problem}
Prove that:
\[
G(K, K^{\prime\prime})
\leq
G(K, K^{\prime})
+
G(K^\prime, K^{\prime\prime}).
\]
Open problem:
\[
u(K\# K^\prime) = u(K) + u(K^\prime).
\]
\end{problem}
\begin{lemma}[Scharlemann '84]
Unknotting number one knots are prime.
\end{lemma}
\subsection*{Tools to bound unknotting number}
\begin{theorem}
For any symmetric polynomial $\Delta \in \mathbb{Z}[t, t^{-1}]$ such that $\Delta(1) = 1$, there exists a knot $K$ such that:
\begin{enumerate}
\item
$K$ has unknotting number $1$,
\item
$\Delta_K = \Delta$.
\end{enumerate}
\end{theorem}
Let us consider a knot $K$ and its Seifert surface $\Sigma$.
the Seifert form for $K_-$
\\
the Seifert form for $K_+$
\\
$S_- + S_+$ differs from
by a term in the bottom right corner
Let $A$ be a symmetric $n \times n$ matrix over $\mathbb{R}$. Let $A_1, \dots, A_n$ be minors of $A$. \\
Let $\epsilon_0 = 1$
If

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@ -98,6 +98,10 @@
\DeclareMathOperator{\sign}{sign} \DeclareMathOperator{\sign}{sign}
\DeclareMathOperator{\odd}{odd} \DeclareMathOperator{\odd}{odd}
\DeclareMathOperator{\even}{even} \DeclareMathOperator{\even}{even}
\DeclareMathOperator{\Diff}{Diff}
\DeclareMathOperator{\Iso}{Iso}
\DeclareMathOperator{\mcg}{MCG}
@ -126,45 +130,92 @@
%\newpage %\newpage
%\input{myNotes} %\input{myNotes}
\section{Basic definitions \hfill\DTMdate{2019-02-25}} \section{Basic definitions
\texorpdfstring{
\hfill \DTMdate{2019-02-25}}
{}}
\input{lec_25_02.tex} \input{lec_25_02.tex}
\section{Alexander polynomial \hfill\DTMdate{2019-03-04}} \section{Alexander polynomial
\texorpdfstring{
\hfill\DTMdate{2019-03-04}}
{}}
\input{lec_04_03.tex} \input{lec_04_03.tex}
%add Hurewicz theorem? %add Hurewicz theorem?
\section{Examples of knot classes \section{Examples of knot classes
\texorpdfstring{
\hfill\DTMdate{2019-03-11}} \hfill\DTMdate{2019-03-11}}
{}}
\input{lec_11_03.tex} \input{lec_11_03.tex}
\section{Concordance group \hfill\DTMdate{2019-03-18}} \section{Concordance group
\texorpdfstring{
\hfill\DTMdate{2019-03-18}}
{}}
\input{lec_18_03.tex} \input{lec_18_03.tex}
\section{Genus $g$ cobordism \hfill\DTMdate{2019-03-25}} \section{Genus
\texorpdfstring{$g$}{g}
cobordism
\texorpdfstring{
\hfill\DTMdate{2019-03-25}}
{}}
\input{lec_25_03.tex} \input{lec_25_03.tex}
\section{\hfill\DTMdate{2019-04-08}} \section{
\texorpdfstring{
\hfill\DTMdate{2019-04-08}}
{}}
\input{lec_08_04.tex} \input{lec_08_04.tex}
\section{Linking form\hfill\DTMdate{2019-04-15}} \section{Linking form
\texorpdfstring{
\hfill\DTMdate{2019-04-15}}
{}}
\input{lec_15_04.tex} \input{lec_15_04.tex}
\section{\hfill\DTMdate{2019-05-06}} \section{
\texorpdfstring{
\hfill\DTMdate{2019-05-06}}
{}}
\input{lec_06_05.tex} \input{lec_06_05.tex}
\section{\hfill\DTMdate{2019-05-20}} % no lecture at 13.05
%\section{\hfill\DTMdate{2019-05-20}}
%\input{lec_13_05.tex}
\section{
\texorpdfstring{
\hfill\DTMdate{2019-05-20}}
{}}
\input{lec_20_05.tex} \input{lec_20_05.tex}
\section{\hfill\DTMdate{2019-05-27}} \section{
\texorpdfstring{
\hfill\DTMdate{2019-05-27}}
{}}
\input{lec_27_05.tex} \input{lec_27_05.tex}
\section{Surgery \hfill\DTMdate{2019-06-03}} \section{
\texorpdfstring{
Surgery \hfill\DTMdate{2019-06-03}}
{}}
\input{lec_03_06.tex} \input{lec_03_06.tex}
\section{Surgery\hfill\DTMdate{2019-06-03}} \section{Surgery
\texorpdfstring{
\hfill\DTMdate{2019-06-10}}
{}}
\input{lec_10_06.tex} \input{lec_10_06.tex}
\section{Mess
\texorpdfstring{
\hfill\DTMdate{2019-06-17}}
{}}
\input{mess.tex}
\end{document} \end{document}