diff --git a/images/concordance_sum.pdf b/images/concordance_sum.pdf new file mode 100644 index 0000000..6922a7d Binary files /dev/null and b/images/concordance_sum.pdf differ diff --git a/images/concordance_sum.pdf_tex b/images/concordance_sum.pdf_tex new file mode 100644 index 0000000..d856807 --- /dev/null +++ b/images/concordance_sum.pdf_tex @@ -0,0 +1,68 @@ +%% Creator: Inkscape inkscape 0.92.2, www.inkscape.org +%% PDF/EPS/PS + LaTeX output extension by Johan Engelen, 2010 +%% Accompanies image file 'concordance_sum.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}{2270.54270346bp}% + \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}(1,0.35664263)% + \put(0.34523208,0.32026902){\color[rgb]{0,0,0}\makebox(0,0)[lt]{\begin{minipage}{0.13180612\unitlength}\raggedright \end{minipage}}}% + \put(0.37403638,0.28673265){\color[rgb]{0,0,0}\makebox(0,0)[lt]{\begin{minipage}{0.05553241\unitlength}\raggedright \end{minipage}}}% + \put(-0.24570519,0.33337987){\color[rgb]{0,0,0}\makebox(0,0)[lt]{\begin{minipage}{0.07663471\unitlength}\raggedright \end{minipage}}}% + \put(0,0){\includegraphics[width=\unitlength,page=1]{concordance_sum.pdf}}% + \put(0.00755836,0.35865552){\color[rgb]{0,0,0}\makebox(0,0)[lt]{\begin{minipage}{0.03301039\unitlength}\raggedright $K_1$\end{minipage}}}% + \put(0.31573503,0.30921292){\color[rgb]{0,0,0}\makebox(0,0)[lt]{\begin{minipage}{0.11352441\unitlength}\raggedright $K_1\prime$\end{minipage}}}% + \put(0.11757955,0.35187197){\color[rgb]{0,0,0}\makebox(0,0)[lt]{\begin{minipage}{0.15898166\unitlength}\raggedright \shortstack{Annulus $A_1$}\end{minipage}}}% + \put(0.1209483,0.44515339){\color[rgb]{0,0,0}\makebox(0,0)[lt]{\begin{minipage}{0.11151923\unitlength}\raggedright \end{minipage}}}% + \put(0,0){\includegraphics[width=\unitlength,page=2]{concordance_sum.pdf}}% + \put(0.00912905,0.16388952){\color[rgb]{0,0,0}\makebox(0,0)[lt]{\begin{minipage}{0.03301039\unitlength}\raggedright $K_2$\end{minipage}}}% + \put(0.30238413,0.11444693){\color[rgb]{0,0,0}\makebox(0,0)[lt]{\begin{minipage}{0.11352441\unitlength}\raggedright $K_2\prime$\end{minipage}}}% + \put(0.11915025,0.15710597){\color[rgb]{0,0,0}\makebox(0,0)[lt]{\begin{minipage}{0.16683513\unitlength}\raggedright \shortstack{Annulus $A_2$}\end{minipage}}}% + \put(0,0){\includegraphics[width=\unitlength,page=3]{concordance_sum.pdf}}% + \put(0.79479654,0.15057288){\color[rgb]{0,0,0}\makebox(0,0)[lt]{\begin{minipage}{0.21876087\unitlength}\raggedright $K_1\prime \# K_2\prime$\end{minipage}}}% + \put(0.39316361,0.18038181){\color[rgb]{0,0,0}\makebox(0,0)[lt]{\begin{minipage}{0.07306307\unitlength}\raggedright $K_1 \# K_2$\end{minipage}}}% + \end{picture}% +\endgroup% diff --git a/images/concordance_sum.svg b/images/concordance_sum.svg new file mode 100644 index 0000000..abb8310 --- /dev/null +++ b/images/concordance_sum.svg @@ -0,0 +1,469 @@ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + image/svg+xml + + + + + + + + + + + + $K_1$ $K_1\prime$ \shortstack{Annulus $A_1$} + + + + + + + $K_2$ $K_2\prime$ \shortstack{Annulus $A_2$} + + + + + $K_1\prime \# K_2\prime$ $K_1 \# K_2$ + diff --git a/lectures_on_knot_theory.tex b/lectures_on_knot_theory.tex index d759edf..2328b94 100644 --- a/lectures_on_knot_theory.tex +++ b/lectures_on_knot_theory.tex @@ -225,7 +225,7 @@ We smooth all the crossings, so we get a disjoint union of circles on the plane. \end{figure} \noindent -Note: in general the obtained surface 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 $D_2$; now 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 $D_2$; now we glue both components on the boundaries: $\partial D_1$ and $\partial D_2$. \begin{figure}[h] \begin{center} @@ -295,6 +295,13 @@ $T(6, 2)$ link: \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 $\Sigma$. +\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 $\alpha_1, \dots, \alpha_n$. @@ -583,7 +590,9 @@ An oriented knot is called negative amphichiral if the mirror image $m(K)$ if $K Prove that if $K$ is negative amphichiral, then $K \# K$ in $\mathbf{C}$ \end{example} - +% +% +% \section{\hfill\DTMdate{2019-03-18}} \begin{definition} A knot $K$ is called (smoothly) slice if $K$ is smoothly concordant to an unknot. \\ @@ -612,9 +621,19 @@ For any $K$, $K \# m(K)$ is slice. \begin{fact} Concordance is an equivalence relation. \end{fact} -\begin{fact} +\begin{fact}\label{fakt:concordance_connected} If $K_1 \sim {K_1}^{\prime}$ and $K_2 \sim {K_2}^{\prime}$, then $K_1 \# K_2 \sim {K_1}^{\prime} \# {K_2}^{\prime}$. +\begin{figure}[h] +\fontsize{10}{10}\selectfont +\centering{ +\def\svgwidth{\linewidth} +\resizebox{0.8\textwidth}{!}{\input{images/concordance_sum.pdf_tex}} +} +\caption{Sketch for Fakt \ref{fakt:concordance_connected}.} +\label{fig:concordance_sum} +\end{figure} + \end{fact} \begin{fact} $K \# m(K) \sim $ the unknot. @@ -649,6 +668,13 @@ $A \cdot B$ doesn't depend of choice of $A$ and $B$ in their homology classes. \end{proposition} + +\section{\hfill\DTMdate{2019-03-11}} +\begin{definition} +A link $L$ is fibered if there exists a map ${\phi: S^3\setminus L \longleftarrow S^1}$ which is locally trivial fibration. +\end{definition} + + \section{\hfill\DTMdate{2019-04-15}} In other words:\\ Choose a basis $(b_1, ..., b_i)$ \\