correction of first lecture starts

2019-10-25 07:02:17 +02:00 · 2019-10-25 07:02:17 +02:00 · f257b26d0a
commit f257b26d0a
parent 26f0821ce5
18 changed files with 1403 additions and 82 deletions
--- a/images/dehn_twist.pdf
+++ b/images/dehn_twist.pdf
--- a/images/dehn_twist.svg
+++ b/images/dehn_twist.svg
@ -20,6 +20,18 @@
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--- a/images/seifert_alg.pdf_tex
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--- a/images/torus_lambda.pdf
+++ b/images/torus_lambda.pdf
--- a/images/torus_lambda.pdf_tex
+++ b/images/torus_lambda.pdf_tex
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--- a/images/torus_lambda.svg
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--- a/images/torus_mu_lambda.pdf
+++ b/images/torus_mu_lambda.pdf
--- a/images/torus_mu_lambda.pdf_tex
+++ b/images/torus_mu_lambda.pdf_tex
@ -0,0 +1,61 @@
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+    \put(0.80750641,0.20322471){\color[rgb]{0,0,0}\makebox(0,0)[lt]{\begin{minipage}{0.14685617\unitlength}\raggedright $K$\end{minipage}}}%
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+    \put(0.66285955,0.08408629){\color[rgb]{0,0,0}\makebox(0,0)[lt]{\begin{minipage}{0.38612036\unitlength}\raggedright $N(K) = D^2 \times S^1$\end{minipage}}}%
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--- a/images/torus_mu_lambda.svg
+++ b/images/torus_mu_lambda.svg
--- a/lec_03_06.tex
+++ b/lec_03_06.tex
@ -69,5 +69,31 @@ S^1 \times \pt  &\longrightarrow \pt \times S^1 \\
 }
 \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 $H_1(S^3 \setminus N, \mathbb{Z})$.


--- a/lec_04_03.tex
+++ b/lec_04_03.tex
@ -1,4 +1,4 @@
-\subsection{Existence of Seifert surface - second proof}
+\subsection{Existence of a Seifert surface - second proof}
 %\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}
@ -209,7 +209,8 @@ There are not trivial knots with Alexander polynomial equal $1$, for example:
 $\Delta_{11n34} \equiv 1$.
 \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:
 \[
 S^3 = \partial D^4 = \partial (D^2 \times D^2) =  (D^2 \times S^1) \cup (S^1 \times D^2).
--- a/lec_06_05.tex
+++ b/lec_06_05.tex
@ -17,12 +17,17 @@ The infinite cyclic cover of a knot complement  $X$ is the cover associated with
 \label{fig:covering}
 }
 \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]
 \fontsize{10}{10}\selectfont
 \centering{
 \def\svgwidth{\linewidth}
 \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}
 }
 \end{figure}
--- a/lec_08_04.tex
+++ b/lec_08_04.tex
@ -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). 

-\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})
-\end{align*}
+\[
+H_2(X, \mathbb{Z}) \xrightarrow{\text{Poincar\'e duality}} H^2(X, \mathbb{Z} ) \xrightarrow{\text{evaluation}}\Hom(H_2(X, \mathbb{Z}), \mathbb{Z}).
+\]
 \noindent
 Intersection form: 
 $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}}
 }
 \caption{$T_X A + T_X B = T_X X$
-}\label{fig:torus_alpha_beta}
+}\label{fig:intersection}
 \end{figure}
 ???????????????????????
 \begin{align*}
--- a/lec_10_06.tex
+++ b/lec_10_06.tex
@ -1,48 +1 @@
-
-
-
-
-\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}
+Consider a surgery
--- a/lec_20_05.tex
+++ b/lec_20_05.tex
@ -60,7 +60,7 @@ H_1(\widetilde{X}, \mathbb{Z}[t, t^{-1}]) \longrightarrow \quot{\mathbb{Q}}{\mat
 \begin{align*}
 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}]}\\
-(\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{fact}
 \noindent
--- a/lec_25_02.tex
+++ b/lec_25_02.tex
@ -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}
 \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}
+\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}
 \item 
 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.
 \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 $S^1$.
 \end{definition}
+
 \begin{definition}
 A link with k - components is a (smooth) embedding of $\overbrace{S^1 \sqcup \ldots \sqcup S^1}^k$ in $S^3$.
 \end{definition}
+
 \begin{example}
 Links:
 \begin{itemize}
--- a/lec_27_05.tex
+++ b/lec_27_05.tex
@ -62,7 +62,80 @@ H_1(\Sigma(K), \mathbb{Z})
 (a, b) \mapsto a{(V + V^T)}^{-1} b
 \end{eqnarray*}
 ???????????????????\\
+???????????????????\\
 \begin{eqnarray*}
 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*}
+(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  
--- a/lectures_on_knot_theory.tex
+++ b/lectures_on_knot_theory.tex
@ -98,6 +98,10 @@
 \DeclareMathOperator{\sign}{sign}
 \DeclareMathOperator{\odd}{odd}
 \DeclareMathOperator{\even}{even}
+\DeclareMathOperator{\Diff}{Diff}
+\DeclareMathOperator{\Iso}{Iso}
+\DeclareMathOperator{\mcg}{MCG}
+



@ -126,45 +130,92 @@
 %\newpage
 %\input{myNotes}

-\section{Basic definitions \hfill\DTMdate{2019-02-25}}
+\section{Basic definitions 
+\texorpdfstring{
+\hfill \DTMdate{2019-02-25}}
+{}}
 \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}
 %add Hurewicz theorem?


 \section{Examples of knot classes
+\texorpdfstring{
 \hfill\DTMdate{2019-03-11}}
+{}}
 \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}

-\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}

-\section{\hfill\DTMdate{2019-04-08}}
+\section{
+\texorpdfstring{
+\hfill\DTMdate{2019-04-08}}
+{}}
 \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}

-\section{\hfill\DTMdate{2019-05-06}}
+\section{
+\texorpdfstring{
+\hfill\DTMdate{2019-05-06}}
+{}}
 \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}

-\section{\hfill\DTMdate{2019-05-27}}
+\section{
+\texorpdfstring{
+\hfill\DTMdate{2019-05-27}}
+{}}
 \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}

-\section{Surgery\hfill\DTMdate{2019-06-03}}
+\section{Surgery
+\texorpdfstring{
+\hfill\DTMdate{2019-06-10}}
+{}}
 \input{lec_10_06.tex}

+\section{Mess
+\texorpdfstring{
+\hfill\DTMdate{2019-06-17}}
+{}}
+\input{mess.tex}
+
 \end{document}