diff --git a/images/dehn_twist.pdf b/images/dehn_twist.pdf index 44a5e65..f518c60 100644 Binary files a/images/dehn_twist.pdf and b/images/dehn_twist.pdf differ diff --git a/images/dehn_twist.svg b/images/dehn_twist.svg index 047c40e..d0c3aa0 100644 --- a/images/dehn_twist.svg +++ b/images/dehn_twist.svg @@ -20,6 +20,18 @@ sodipodi:docname="dehn_twist.svg"> + + + + + + d="m 111.74936,64.635353 c -1.50857,8.628051 -9.97194,8.071102 -15.333563,10.057856 -7.832514,2.902346 -6.903247,10.409815 -7.997307,8.795788 -3.588928,2.753658 2.386444,-14.418694 -8.570427,-12.985964 -5.319088,0.695529 -4.040443,4.850942 -8.395047,4.559546 -4.743437,-0.305985 -7.689258,-7.475306 -7.689258,-10.427226 0,-2.95192 2.685486,-5.624383 7.027329,-7.558865 4.341844,-1.934483 10.340044,-3.130985 16.965474,-3.130985 6.625429,0 12.623629,1.196502 16.965469,3.130985 4.34184,1.934482 7.02733,4.606945 7.02733,7.558865 z" + style="opacity:1;fill:none;fill-opacity:1;fill-rule:evenodd;stroke:url(#radialGradient1721);stroke-width:1.06069636;stroke-linejoin:bevel;stroke-miterlimit:4;stroke-dasharray:none;stroke-dashoffset:0;stroke-opacity:0.97073173" /> image/svg+xml - + diff --git a/images/torus_mu_lambda.pdf b/images/torus_mu_lambda.pdf new file mode 100644 index 0000000..25675b1 Binary files /dev/null and b/images/torus_mu_lambda.pdf differ diff --git a/images/torus_mu_lambda.pdf_tex b/images/torus_mu_lambda.pdf_tex new file mode 100644 index 0000000..0b01573 --- /dev/null +++ b/images/torus_mu_lambda.pdf_tex @@ -0,0 +1,61 @@ +%% Creator: Inkscape inkscape 0.92.2, www.inkscape.org +%% PDF/EPS/PS + LaTeX output extension by Johan Engelen, 2010 +%% Accompanies image file 'torus_mu_lambda.pdf' (pdf, eps, ps) +%% +%% To include the image in your LaTeX document, write +%% \input{.pdf_tex} +%% instead of +%% \includegraphics{.pdf} +%% To scale the image, write +%% \def\svgwidth{} +%% \input{.pdf_tex} +%% instead of +%% \includegraphics[width=]{.pdf} +%% +%% Images with a different path to the parent latex file can +%% be accessed with the `import' package (which may need to be +%% installed) using +%% \usepackage{import} +%% in the preamble, and then including the image with +%% \import{}{.pdf_tex} +%% Alternatively, one can specify +%% \graphicspath{{/}} +%% +%% For more information, please see info/svg-inkscape on CTAN: +%% http://tug.ctan.org/tex-archive/info/svg-inkscape +%% +\begingroup% + \makeatletter% + \providecommand\color[2][]{% + \errmessage{(Inkscape) Color is used for the text in Inkscape, but the package 'color.sty' is not loaded}% + \renewcommand\color[2][]{}% + }% + \providecommand\transparent[1]{% + \errmessage{(Inkscape) Transparency is used (non-zero) for the text in Inkscape, but the package 'transparent.sty' is not loaded}% + \renewcommand\transparent[1]{}% + }% + \providecommand\rotatebox[2]{#2}% + \ifx\svgwidth\undefined% + \setlength{\unitlength}{185.50670233bp}% + \ifx\svgscale\undefined% + \relax% + \else% + \setlength{\unitlength}{\unitlength * \real{\svgscale}}% + \fi% + \else% + \setlength{\unitlength}{\svgwidth}% + \fi% + \global\let\svgwidth\undefined% + \global\let\svgscale\undefined% + \makeatother% + \begin{picture}(1,0.62539361)% + \put(1.05585685,2.21258294){\color[rgb]{0,0,0}\makebox(0,0)[lt]{\begin{minipage}{0.51818377\unitlength}\raggedright \end{minipage}}}% + \put(0,0){\includegraphics[width=\unitlength,page=1]{torus_mu_lambda.pdf}}% + \put(0.89358894,0.61469018){\color[rgb]{0,0,0}\makebox(0,0)[lt]{\begin{minipage}{0.10667827\unitlength}\raggedright $\lambda$\end{minipage}}}% + \put(0.01370736,0.19722543){\color[rgb]{0,0,0}\makebox(0,0)[lt]{\begin{minipage}{0.15987467\unitlength}\raggedright $\mu$\end{minipage}}}% + \put(0,0){\includegraphics[width=\unitlength,page=2]{torus_mu_lambda.pdf}}% + \put(0.80750641,0.20322471){\color[rgb]{0,0,0}\makebox(0,0)[lt]{\begin{minipage}{0.14685617\unitlength}\raggedright $K$\end{minipage}}}% + \put(0,0){\includegraphics[width=\unitlength,page=3]{torus_mu_lambda.pdf}}% + \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}}}% + \end{picture}% +\endgroup% diff --git a/images/torus_mu_lambda.svg b/images/torus_mu_lambda.svg new file mode 100644 index 0000000..79eab56 --- /dev/null +++ b/images/torus_mu_lambda.svg @@ -0,0 +1,1109 @@ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + image/svg+xml + + + + + + + $\Sigma$ + + + $\lambda$ $\mu$ + + $K$ + $N(K) = D^2 \times S^1$ + diff --git a/lec_03_06.tex b/lec_03_06.tex index 669340b..5ae620f 100644 --- 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 $H_1(S^3 \setminus N, \mathbb{Z})$. diff --git a/lec_04_03.tex b/lec_04_03.tex index a76580f..eeb7dda 100644 --- 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). diff --git a/lec_06_05.tex b/lec_06_05.tex index 8af4b6a..601c6b4 100644 --- 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} diff --git a/lec_08_04.tex b/lec_08_04.tex index e432602..4dce8d2 100644 --- 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*} diff --git a/lec_10_06.tex b/lec_10_06.tex index 9541a40..1b1f8a4 100644 --- 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 diff --git a/lec_20_05.tex b/lec_20_05.tex index 020911b..70cf05d 100644 --- 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 diff --git a/lec_25_02.tex b/lec_25_02.tex index 7586384..3e1c85b 100644 --- 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} diff --git a/lec_27_05.tex b/lec_27_05.tex index 1b7b240..a18c596 100644 --- 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 \ No newline at end of file diff --git a/lectures_on_knot_theory.tex b/lectures_on_knot_theory.tex index be651af..097d940 100644 --- 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}