diff --git a/images/covering.pdf_tex b/images/covering.pdf_tex index fab7845..ae66cee 100644 --- a/images/covering.pdf_tex +++ b/images/covering.pdf_tex @@ -58,13 +58,13 @@ \put(0.80142758,0.16860343){\color[rgb]{0,0,0}\makebox(0,0)[lt]{\begin{minipage}{0.08408535\unitlength}\raggedright $N_1$\end{minipage}}}% \put(0.21025104,0.17098059){\color[rgb]{0,0,0}\makebox(0,0)[lt]{\begin{minipage}{0.08408535\unitlength}\raggedright $N_{-1}$\end{minipage}}}% \put(0,0){\includegraphics[width=\unitlength,page=2]{covering.pdf}}% - \put(0.06101596,0.07198974){\color[rgb]{0,0,0}\makebox(0,0)[lt]{\begin{minipage}{0.18711552\unitlength}\raggedright \shortstack{$ t^{-1}\alpha_1, t^{-1}\alpha_2, \cdots, t^{-1}\alpha_n$}\end{minipage}}}% - \put(0.56677018,0.35673072){\color[rgb]{0,0,0}\makebox(0,0)[lt]{\begin{minipage}{0.30643553\unitlength}\raggedright \shortstack{$ta_1, ta_2, \cdots, ta_n$}\end{minipage}}}% - \put(0.28338509,0.3553748){\color[rgb]{0,0,0}\makebox(0,0)[lt]{\begin{minipage}{0.21016596\unitlength}\raggedright $a_1, a_2, \cdots, a_n$\end{minipage}}}% + \put(0.06101596,0.07198974){\color[rgb]{0,0,0}\makebox(0,0)[lt]{\begin{minipage}{0.18711552\unitlength}\raggedright \shortstack{$ t^{-1}\alpha_1, t^{-1}\alpha_2, \dots, t^{-1}\alpha_n$}\end{minipage}}}% + \put(0.56677018,0.35673072){\color[rgb]{0,0,0}\makebox(0,0)[lt]{\begin{minipage}{0.30643553\unitlength}\raggedright \shortstack{$ta_1, ta_2, \dots, ta_n$}\end{minipage}}}% + \put(0.28338509,0.3553748){\color[rgb]{0,0,0}\makebox(0,0)[lt]{\begin{minipage}{0.21016596\unitlength}\raggedright $a_1, a_2, \dots, a_n$\end{minipage}}}% \put(0.27524963,0.02317699){\color[rgb]{0,0,0}\makebox(0,0)[lt]{\begin{minipage}{0.21016596\unitlength}\raggedright $a_j$\end{minipage}}}% \put(0.41626421,0.02317699){\color[rgb]{0,0,0}\makebox(0,0)[lt]{\begin{minipage}{0.21016596\unitlength}\raggedright $a_i^+$\end{minipage}}}% \put(0,0){\includegraphics[width=\unitlength,page=3]{covering.pdf}}% - \put(0.48541563,0.07605746){\color[rgb]{0,0,0}\makebox(0,0)[lt]{\begin{minipage}{0.37829873\unitlength}\raggedright \shortstack{$\alpha_1, \alpha_2, \cdots, \alpha_n$}\end{minipage}}}% + \put(0.48541563,0.07605746){\color[rgb]{0,0,0}\makebox(0,0)[lt]{\begin{minipage}{0.37829873\unitlength}\raggedright \shortstack{$\alpha_1, \alpha_2, \dots, \alpha_n$}\end{minipage}}}% \put(0,0){\includegraphics[width=\unitlength,page=4]{covering.pdf}}% \end{picture}% \endgroup% diff --git a/images/covering.svg b/images/covering.svg index ff952bc..c0ac593 100644 --- a/images/covering.svg +++ b/images/covering.svg @@ -106,12 +106,12 @@ inkscape:pageopacity="0.0" inkscape:pageshadow="2" inkscape:zoom="0.24748737" - inkscape:cx="358.42434" + inkscape:cx="1328.1708" inkscape:cy="556.20482" inkscape:document-units="mm" inkscape:current-layer="layer1" showgrid="false" - inkscape:window-width="1395" + inkscape:window-width="1399" inkscape:window-height="855" inkscape:window-x="0" inkscape:window-y="1" @@ -299,7 +299,7 @@ height="157.58377" width="557.60419" id="rect3965" />\shortstack{$ t^{-1}\alpha_1, t^{-1}\alpha_2, \cdots, t^{-1}\alpha_n$} \shortstack{$ t^{-1}\alpha_1, t^{-1}\alpha_2, \dots, t^{-1}\alpha_n$} \shortstack{$ta_1, ta_2, \cdots, ta_n$} \shortstack{$ta_1, ta_2, \dots, ta_n$} $a_1, a_2, \cdots, a_n$ $a_1, a_2, \dots, a_n$ \shortstack{$\alpha_1, \alpha_2, \cdots, \alpha_n$} \shortstack{$\alpha_1, \alpha_2, \dots, \alpha_n$} \shortstack{$a_1, \cdots, a_n$ - generators of $H_1(S)$} \shortstack{$a_1, \dots, a_n$ - generators of $H_1(S)$} \shortstack{$\alpha_1, \cdots, \alpha_n$ - dual generators of $H_1(N)$} \shortstack{$\alpha_1, \dots, \alpha_n$ - dual generators of $H_1(N)$} + id="flowPara5107" /> diff --git a/lectures_on_knot_theory.pdf b/lectures_on_knot_theory.pdf index ff7884a..f97d7be 100644 Binary files a/lectures_on_knot_theory.pdf and b/lectures_on_knot_theory.pdf differ diff --git a/lectures_on_knot_theory.tex b/lectures_on_knot_theory.tex index 916769e..5f57f3f 100644 --- a/lectures_on_knot_theory.tex +++ b/lectures_on_knot_theory.tex @@ -69,7 +69,8 @@ \newcommand{\sdots}{\smash{\vdots}} - +\DeclareRobustCommand\longtwoheadrightarrow + {\relbar\joinrel\twoheadrightarrow} @@ -79,8 +80,11 @@ \DeclareMathOperator{\Hom}{Hom} \DeclareMathOperator{\rank}{rank} -\DeclareMathOperator{\Gl}{Gl} +\DeclareMathOperator{\ord}{ord} +\DeclareMathOperator{\Gl}{GL} +\DeclareMathOperator{\Sl}{SL} \DeclareMathOperator{\Lk}{lk} +\DeclareMathOperator{\pt}{\{pt\}} \titleformat{\section}{\normalfont \fontsize{12}{15} \bfseries}{% @@ -562,7 +566,7 @@ If $K$ is a knot, then $n$ is necessarily even, and so $\Delta_K(t^{-1}) = t^{-n \begin{lemma} \begin{align*} \frac{1}{2} \deg \Delta_K(t) \leq g_3(K), -\text{ where } deg (a_n t^n + \cdots + a_1 t^l )= k - l. +\text{ where } deg (a_n t^n + \dots + a_1 t^l )= k - l. \end{align*} \end{lemma} \begin{proof} @@ -622,12 +626,10 @@ Figure 8 knot is negative amphichiral. % \section{Concordance group \hfill\DTMdate{2019-03-18}} \begin{definition} -A knot $K$ is called (smoothly) slice if $K$ is smoothly concordant to an unknot. \\ -A knot $K$ is smoothly slice if and only if $K$ bounds a smoothly embedded disk in $B^4$. -\end{definition} -\begin{definition} Two knots $K$ and $K^{\prime}$ are called (smoothly) concordant if there exists an annulus $A$ that is smoothly embedded in ${S^3 \times [0, 1]}$ such that -${\partial A = K^{\prime} \times \{1\} \; \sqcup \; K \times \{0\}}$. +\[ +\partial A = K^{\prime} \times \{1\} \; \sqcup \; K \times \{0\}. +\] \end{definition} \begin{figure}[h] @@ -637,6 +639,10 @@ ${\partial A = K^{\prime} \times \{1\} \; \sqcup \; K \times \{0\}}$. \resizebox{0.8\textwidth}{!}{\input{images/concordance.pdf_tex}} } \end{figure} +\begin{definition} +A knot $K$ is called (smoothly) slice if $K$ is smoothly concordant to an unknot. \\ +A knot $K$ is smoothly slice if and only if $K$ bounds a smoothly embedded disk in $B^4$. +\end{definition} @@ -681,9 +687,9 @@ Remark: $K \sim K^{\prime} \Leftrightarrow K \# -K^{\prime}$ is slice. \\ \\ \noindent -Let $\Omega$ be an oriented \\ +Let $\Omega$ be an oriented four-manifold. \\ ???????\\ -Suppose $\Sigma$ is a Seifert matrix with an intersection form ${(\alpha, \beta) \mapsto \Lk(\alpha, \beta^+)}$. Suppose $\alpha, \beta \in H_1(\Sigma, \mathbb{Z}$ (i.e. there are cycles). \\ +Suppose $\Sigma$ is a Seifert surface and $V$ a Seifert form defined on $\Sigma$: ${(\alpha, \beta) \mapsto \Lk(\alpha, \beta^+)}$. Suppose $\alpha, \beta \in H_1(\Sigma, \mathbb{Z})$ (i.e. there are cycles). \\ ??????????????\\ $\alpha, \beta \in \ker (H_1(\Sigma, \mathbb{Z}) \longrightarrow H_1(\Omega, \mathbb{Z}))$. Then there are two cycles $A, B \in \Omega$ such that $\partial A = \alpha$ and $\partial B = \beta$. Let $B^+$ be a push off of $B$ in the positive normal direction such that @@ -692,6 +698,7 @@ Then $\Lk(\alpha, \beta^+) = A \cdot B^+$ % % +%ball_4_alpha_beta.pdf \\ \section{\hfill\DTMdate{2019-04-08}} % @@ -727,7 +734,7 @@ of $H_2(Y, \mathbb{Z}$, then $A = (b_i, b_y)$ \\??\\ is a matrix of intersection \begin{align*} \quot{\mathbb{Z}^n}{A\mathbb{Z}^n} \cong H_1(Y, \mathbb{Z}). \end{align*} -In particular $\mid \det A\mid = \# H_1(Y, \mathbb{Z}$.\\ +In particular $\mid \det A\mid = \# H_1(Y, \mathbb{Z})$.\\ That means - what is happening on boundary is a measure of degeneracy. \begin{center} @@ -750,16 +757,17 @@ H_1(Y, \mathbb{Z}) & \end{tikzcd} $(a, b) \mapsto aA^{-1}b^T$ \end{center} - +?????????????????????????????????\\ +\noindent The intersection form on a four-manifold determines the linking on the boundary. \\ \noindent Let $K \in S^1$ be a knot, $\Sigma(K)$ its double branched cover. If $V$ is a Seifert matrix for $K$, then $H_1(\Sigma(K), \mathbb{Z}) \cong \quot{\mathbb{Z}^n}{A\mathbb{Z}}$ where -$A = V \times V^T$, where $n = \rank V$. +$A = V \times V^T$, $n = \rank V$. %\input{ink_diag} \begin{figure}[h] -\fontsize{40}{10}\selectfont +\fontsize{20}{10}\selectfont \centering{ \def\svgwidth{\linewidth} \resizebox{0.5\textwidth}{!}{\input{images/ball_4.pdf_tex}} @@ -777,20 +785,33 @@ Let $X$ be the four-manifold obtained via the double branched cover of $B^4$ bra \item The intersection form on $X$ is $V + V^T$. \end{itemize} \end{fact} +\begin{figure}[h] +\fontsize{20}{10}\selectfont +\centering{ +\def\svgwidth{\linewidth} +\resizebox{0.5\textwidth}{!}{\input{images/ball_4_pushed_cycle.pdf_tex}} +\caption{Cycle pushed in 4-ball.} +\label{fig:pushCycle} +} +\end{figure} \noindent Let $Y = \Sigma(K)$. Then: -\begin{flalign*} -H_1(Y, \mathbb{Z}) \times H_1(Y, \mathbb{Z}) \longrightarrow \quot{\mathbb{Q}}{\mathbb{Z}}& +\begin{align*} +H_1(Y, \mathbb{Z}) \times H_1(Y, \mathbb{Z}) &\longrightarrow \quot{\mathbb{Q}}{\mathbb{Z}} \\ -(a,b) \mapsto a A^{-1} b^{T},\qquad -A = V + V^T& -\\ -H_1(Y, \mathbb{Z}) \cong \quot{\mathbb{Z}^n}{A\mathbb{Z}}&\\ -A \longrightarrow BAC^T \quad \text{Smith normal form}& -\end{flalign*} +(a,b) &\mapsto a A^{-1} b^{T},\qquad +A = V + V^T. +\end{align*} +???????????????????????????? +\begin{align*} +H_1(Y, \mathbb{Z}) \cong \quot{\mathbb{Z}^n}{A\mathbb{Z}}\\ +A \longrightarrow BAC^T \quad \text{Smith normal form} +\end{align*} ???????????????????????\\ In general +%no lecture at 29.04 + \section{\hfill\DTMdate{2019-05-20}} Let $M$ be compact, oriented, connected four-dimensional manifold. If ${H_1(M, \mathbb{Z}) = 0}$ then there exists a @@ -1040,6 +1061,7 @@ $2 \sum\limits_{k_i \text{ odd}} \epsilon_i$. The peak of the signature function \end{theorem} \end{proof} \section{\hfill\DTMdate{2019-05-27}} + .... \begin{definition} A square hermitian matrix $A$ of size $n$. @@ -1063,8 +1085,12 @@ Remove from $\Delta$ the two self intersecting and glue the Seifert surface for \begin{example} The knot $8_{20}$ is slice: $\sigma \equiv 0$ almost everywhere but $\sigma(e^{\frac{ 2\pi i}{6}}) = + 1$. \end{example} -\subsection{Surgery} -Recall that $H_1(S^1 \times S^1, \mathbb{Z}) = \mathbb{Z}^3$. As generators for $H_1$ we can set ${\alpha = [S^1 \times \{pt\}]}$ and ${\beta=[\{pt\} \times S^1]}$. Suppose ${\phi: S^1 \times S^1 \longrightarrow S^1 \times S^1}$ is a diffeomorphism. +%ref Structure in the classical knot concordance group +%Tim D. Cochran, Kent E. Orr, Peter Teichner +%Journal-ref: Comment. Math. Helv. 79 (2004) 105-123 +\subsection*{Surgery} +%Rolfsen, geometric group theory, Diffeomorpphism of a torus, Mapping class group +Recall that $H_1(S^1 \times S^1, \mathbb{Z}) = \mathbb{Z}^3$. As generators for $H_1$ we can set ${\alpha = [S^1 \times \pt]}$ and ${\beta=[\pt \times S^1]}$. Suppose ${\phi: S^1 \times S^1 \longrightarrow S^1 \times S^1}$ is a diffeomorphism. Consider an induced map on homology group: \begin{align*} H_1(S^1 \times S^1, \mathbb{Z}) \ni \phi_* (\alpha) &= p\alpha + q \beta, \quad p, q \in \mathbb{Z},\\ @@ -1075,9 +1101,30 @@ H_1(S^1 \times S^1, \mathbb{Z}) \ni \phi_* (\alpha) &= p\alpha + q \beta, \quad r & s \end{pmatrix} \end{align*} - +As $\phi_*$ is diffeomorphis, it must be invertible over $\mathbb{Z}$. Then for a direction preserving diffeomorphism we have $\det \phi_* = 1$. Therefore $\phi_* \in \Sl(2, \mathbb{Z})$. \end{theorem} +\vspace{10cm} +\begin{theorem} +Every such a matrix can be realized as a torus. +\end{theorem} +\begin{proof} +\begin{enumerate}[label={(\Roman*)}] +\item +Geometric reason +\begin{align*} +\phi_t: +S^1 \times S^1 &\longrightarrow S^1 \times S^1 \\ +S^1 \times \pt &\longrightarrow \pt \times S^1 \\ +\pt \times S^1 &\longrightarrow S^1 \times \pt \\ +(x, y) & \mapsto (-y, x) +\end{align*} +\item +\end{enumerate} +\end{proof} + + + \section{balagan} @@ -1128,4 +1175,87 @@ has a subspace of dimension $g_{\Sigma}$ on which it is zero: * & \dots & * & * & \dots & * \end{pmatrix}_{2g_{\Sigma} \times 2g_{\Sigma}} \end{align*} + + +\section{\hfill\DTMdate{2019-05-06}} + +\begin{definition} +Let $X$ be a knot complement. +Then $H_1(X, \mathbb{Z}) \cong \mathbb{Z}$ and there exists an epimorphism +$\pi_1(X) \overset{\phi}\twoheadrightarrow \mathbb{Z}$.\\ +The infinite cyclic cover of a knot complement $X$ is the cover associated with the epimorphism $\phi$. +\[ +\widetilde{X} \longtwoheadrightarrow X +\] +\end{definition} +%Rolfsen, bachalor thesis of Kamila +\begin{figure}[h] +\fontsize{10}{10}\selectfont +\centering{ +\def\svgwidth{\linewidth} +\resizebox{1\textwidth}{!}{\input{images/covering.pdf_tex}} +\caption{Infinite cyclic cover of a knot complement.} +\label{fig:covering} +} +\end{figure} +\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.} +\label{fig:complement} +} +\end{figure} +\noindent +Formal sums $\sum \phi_i(t) a_i + \sum \phi_j(t)\alpha_j$ \\ +finitely generated as a $\mathbb{Z}[t, t^{-1}]$ module. +\\ +Let $v_{ij} = \Lk(a_i, a_j^+)$. Then +$V = \{ v_ij\}_{i, j = 1}^n$ is the Seifert matrix associated to the surface $\Sigma$ and the basis $a_1, \dots, a_n$. Therefore $a_k^+ = \sum_{j} v_{jk} \alpha_j$. Then +$\Lk(a_i, a_k^+)= \Lk(a_k^+, a_i) = \sum_j v_{jk} \Lk(\alpha_j, a_i) = v_{ik}$. +We also notice that $\Lk(a_i, a_j^-) = \Lk(a_i^+, a_j)= v_{ij}$ and +$a_j^- = \sum_k v_{kj} t^{-1} \alpha_j$. +\\ +\noindent +The homology of $\widetilde{X}$ is generated by $a_1, \dots, a_n$ and relations. + +\begin{definition} +The $\mathbb{Z}[t, t^{-1}]$ module $H_1(\widetilde{X})$ is called the Alexander module of knot $K$. +\end{definition} +\noindent +Let $R$ be a PID, $M$ a finitely generated $R$ module. Let us consider +\[ +R^k \overset{A} \longrightarrow R^n \longtwoheadrightarrow M, +\] +where $A$ is a $k \times n$ matrix, assume $k\ge n$. The order of $M$ is the $\gcd$ of all determinants of the $n \times n$ minors of $A$. If $k = n$ then $\ord M = \det A$. +\begin{theorem} +Order of $M$ doesn't depend on $A$. +\end{theorem} +\noindent +For knots the order of the Alexander module is the Alexander polynomial. +\begin{theorem} +\[ +\forall x \in M: (\ord M) x = 0. +\] +\end{theorem} +\noindent +$M$ is well defined up to a unit in $R$. +\subsection*{Blanchfield pairing} +\section{balagan} +\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 + \end{document} +