From 3682a71808bdbffc3622238055b1de9b04d2918b Mon Sep 17 00:00:00 2001 From: Maria Marchwicka Date: Fri, 18 Oct 2019 11:19:20 +0200 Subject: [PATCH] first lecture for Maciej --- lec_03_06.tex | 10 + lec_04_03.tex | 25 +- lec_06_05.tex | 109 +++--- lec_08_04.pdf | Bin 0 -> 24363 bytes lec_08_04.tex | 6 +- lec_10_06.tex | 48 +++ lec_11_03.tex | 2 +- lec_15_04.pdf | Bin 0 -> 22008 bytes lec_15_04.tex | 6 +- lec_18_03.tex | 35 +- lec_20_05.pdf | Bin 0 -> 24259 bytes lec_20_05.tex | 40 ++- lec_25_02.pdf | Bin 0 -> 23695 bytes lec_25_02.tex | 39 ++- lec_25_03.tex | 6 +- lec_27_05.tex | 70 +++- lectures_on_knot_theory.pdf | Bin 669335 -> 777164 bytes lectures_on_knot_theory.tex | 640 ++---------------------------------- 18 files changed, 290 insertions(+), 746 deletions(-) create mode 100644 lec_08_04.pdf create mode 100644 lec_10_06.tex create mode 100644 lec_15_04.pdf create mode 100644 lec_20_05.pdf create mode 100644 lec_25_02.pdf diff --git a/lec_03_06.tex b/lec_03_06.tex index fefe472..669340b 100644 --- a/lec_03_06.tex +++ b/lec_03_06.tex @@ -6,6 +6,16 @@ u(K) \geq g_4(K) \begin{proof} Recall that if $u(K)=u$ then $K$ bounds a disk $\Delta$ with $u$ ordinary double points. \\ +??????????????? +\\ +\begin{eqnarray*} +\chi (D^2) = 1 \\ +\chi (\Delta) = 1 - u\\ +\gamma = 0 \in \pi_1(B^4 \setminus S) +\end{eqnarray*} + +?????????????? +\\ \noindent Remove from $\Delta$ the two self intersecting disks and glue the Seifert surface for the Hopf link. The reality surface $S$ has Euler characteristic $\chi(S) = 1 - 2u$. Therefore $g_4(S) = u$. \end{proof} diff --git a/lec_04_03.tex b/lec_04_03.tex index 1519178..a76580f 100644 --- a/lec_04_03.tex +++ b/lec_04_03.tex @@ -41,16 +41,19 @@ Let us consider a long exact sequence of cohomology of a pair $(S^3, S^3 \setmin & \mathbb{Z} \ar[u,isomorphic] &\\ \end{tikzcd} \end{center} +The tubular neighbourhood of the knot is homomorphic to +$D^2 \times S^1$. +So its boundary +$\partial N \cong \ S^1 \times S^1$ and therefore: +$H^1(N, \partial N) \cong \ \mathbb{Z} \oplus \mathbb{Z}$. By excision theorem we have: \begin{align*} -N \cong & D^2 \times S^1\\ -\partial N \cong & S^1 \times S^1\\ -H^1(N, \partial N) \cong & \mathbb{Z} \oplus \mathbb{Z} +H^* (S^3, S^3 \setminus N) &\cong H^* (N, \partial N). \end{align*} +Therefore: \begin{align*} -H^* (S^3, S^3 \setminus N) &\cong H^* (N, \partial N)\\ -\\ -H^ 1 (S^3\setminus N) &\cong H^1(S^3\setminus K) \cong \mathbb{Z} +H^ 1 (S^3\setminus N) &\cong H^1(S^3\setminus K) \cong \mathbb{Z}. \end{align*} +Let us consider the following diagram: \begin{equation*} \begin{tikzcd}[row sep=huge] H^1(S^3 \setminus K) \arrow[r,] \arrow[d,"\widetilde{\Theta}"] & @@ -62,9 +65,11 @@ H^1(N \setminus K) \arrow[d,"\Theta"] \\ \noindent $\Sigma = \widetilde{\Theta}^{-1}(X)$ is a surface, such that $\partial \Sigma = K$, so it is a Seifert surface. % -% +% picture for excision theorem % Thom isomorphism, \end{proof} +%$S$ - equivalence $\Sigma$\\ +%simple closed curves $\alpha_1, ... \alpha_n \in H_1(\Sigma, \mathbb{Z})$ basis for $H_1$ \subsection{Alexander polynomial} \begin{definition} Let $S$ be a Seifert matrix for a knot $K$. The Alexander polynomial $\Delta_K(t)$ is a Laurent polynomial: @@ -256,13 +261,13 @@ Suppose $K \subset S^3$ and $\pi_1(S^3 \setminus K)$ is infinite cyclic ($\mathb \end{corollary} \begin{proof} Let $N$ be a tubular neighbourhood of a knot $K$ and $M = S^3 \setminus N$ its complement. Then $\partial M = S^1 \times S^1$. Let $f : \pi_1(\partial M ) \longrightarrow \pi_1(M)$. -If $\pi_1(M)$ is infinite cyclic group then the map $f$ is non-trivial. Suppose ${\lambda \in \ker (\pi_1(S^1 \times S^1) \longrightarrow \pi_1(M)}$. +If $\pi_1(M)$ is infinite cyclic group then the map $f$ is non-trivial. Suppose ${\lambda \in \ker (\pi_1(S^1 \times S^1) \longrightarrow \pi_1(M))}$. There is a map $g: (D^2, \partial D^2) \longrightarrow (M, \partial M)$ such that $g(\partial D^2) = \lambda$.\\ By Dehn's lemma there exists an embedding ${h: (D^2, \partial D^2) \longhookrightarrow (M, \partial M)}$ such that $h\big|_{\partial D^2} = f \big|_{\partial D^2}$ and $h(\partial D^2) = \lambda$. Let $\Sigma$ be a union of the annulus and the image of $\partial D^2$. -\\???? $g_3$?\\ -If $g(\Sigma) = 0$, then $K$ is trivial. \\ +%$g_3$ +If $g_3(\Sigma) = 0$, then $K$ is trivial. \\ Now we should proof that: \[ H_1(M) \cong \mathbb{Z} \Longrightarrow \lambda \in \ker ( \pi_1(S^1 \times S^1) \longrightarrow \pi_1(M)). diff --git a/lec_06_05.tex b/lec_06_05.tex index e98986c..8af4b6a 100644 --- a/lec_06_05.tex +++ b/lec_06_05.tex @@ -38,21 +38,6 @@ $a_j^- = \sum_k v_{kj} t^{-1} \alpha_j$. \\ \noindent The homology of $\widetilde{X}$ is generated by $a_1, \dots, a_n$ and relations. -\begin{definition} -The Nakanishi index of a knot is the minimal number of generators of $H_1(\widetilde{X})$. -\end{definition} -%see Maciej page -\noindent -Remark about notation: sometimes one writes $H_1(X; \mathbb{Z}[t, t^{-1}])$ (what is also notation for twisted homology) instead of $H_1(\widetilde{X})$. -\\ -????????????????????? -\\ -\noindent -$\Sigma_?(K) \rightarrow S^3$ ?????\\ -$H_1(\Sigma_?(K), \mathbb{Z}) = h$\\ -$H \times H \longrightarrow \quot{\mathbb{Q}}{\mathbb{Z}}$\\ -...\\ - Let now $H = H_1(\widetilde{X})$. Can we define a paring? \\ Let $c, d \in H(\widetilde{X})$ (see Figure \ref{fig:covering_pairing}), $\Delta$ an Alexander polynomial. We know that $\Delta c = 0 \in H_1(\widetilde{X})$ (Alexander polynomial annihilates all possible elements). Let consider a surface $F$ such that $\partial F = c$. Now consider intersection points $F \cdot d$. This points can exist in any $N_k$ or $S_k$. \[ @@ -60,19 +45,26 @@ Let $c, d \in H(\widetilde{X})$ (see Figure \ref{fig:covering_pairing}), $\Delta \] \\ ?????????????\\ -\begin{figure}[h] -\fontsize{10}{10}\selectfont -\centering{ -\def\svgwidth{\linewidth} -\resizebox{1\textwidth}{!}{\input{images/covering_pairing.pdf_tex}} -\caption{$c, d \in H_1(\widetilde{X})$.} -\label{fig:covering_pairing} -} -\end{figure} - +There is at least one paper where the structure of (Alexander module?) is calculated from a specific knot (?minimal number of generators?) +\\ +C. Kearton, S. M. J. Wilson +\\ +\begin{fact} +Let $A$ be a matrix over principal ideal domain $R$. Than there exist matrices $C$, $D$ and $E$ such that $A = CDE$, +\[D = \begin{bmatrix} +d_1 & 0 & \cdots & \cdots & 0 \\ +0 & d_2 & 0 & \cdots & 0 \\ +\sdots & & \ddots & & \sdots & \\ +0 & \cdots & 0 & d_{n-1} & 0\\ +0 & \cdots & \cdots & 0 & d_n +\end{bmatrix},\] +where $d_{i + 1} | d_i$, and matrices +$C$ and $E$ are invertible over $R$.\\ +$D$ is called a Smith normal form of the matrix $A$. +\end{fact} \begin{definition} -The $\mathbb{Z}[t, t^{-1}]$ module $H_1(\widetilde{X})$ is called the Alexander module of knot $K$. +The $\mathbb{Z}[t, t^{-1}]$ module $H_1(\widetilde{X})$ is called the Alexander module of a knot $K$. \end{definition} \noindent Let $R$ be a PID, $M$ a finitely generated $R$ module. Let us consider @@ -92,53 +84,36 @@ For knots the order of the Alexander module is the Alexander polynomial. \end{theorem} \noindent $M$ is well defined up to a unit in $R$. -\subsection*{Blanchfield pairing} -\section{balagan} - -\begin{fact}[Milnor Singular Points of Complex Hypersurfaces] -\end{fact} -%\end{comment} +\\ +??????????????????\\ +General picture : $K$, $X$ knot complement... +\begin{eqnarray*} +H_1( X, \mathbb{Z}) = \mathbb{Z} \\ +H_1(\widetilde{X}, \mathbb{Z}[t, t^{-1}]) \\ +\pi_1(X) +\end{eqnarray*} +\begin{definition} +The Nakanishi index of a knot is the minimal number of generators of $H_1(\widetilde{X})$. +\end{definition} +%see Maciej page \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} +Remark about notation: sometimes one writes $H_1(X; \mathbb{Z}[t, t^{-1}])$ (what is also notation for twisted homology) instead of $H_1(\widetilde{X})$. +\\ +????????????????????? +\\ \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 - +$\Sigma_?(K) \rightarrow S^3$ ?????\\ +$H_1(\Sigma_?(K), \mathbb{Z}) = h$\\ +$H \times H \longrightarrow \quot{\mathbb{Q}}{\mathbb{Z}}$\\ +...\\ \begin{figure}[h] \fontsize{10}{10}\selectfont \centering{ \def\svgwidth{\linewidth} -\resizebox{0.5\textwidth}{!}{\input{images/ball_4_alpha_beta.pdf_tex}} +\resizebox{1\textwidth}{!}{\input{images/covering_pairing.pdf_tex}} +\caption{$c, d \in H_1(\widetilde{X})$.} +\label{fig:covering_pairing} } -%\caption{Sketch for Fact %%\label{fig:concordance_m} \end{figure} -\end{document} - +\subsection*{Blanchfield pairing} diff --git a/lec_08_04.pdf b/lec_08_04.pdf new file mode 100644 index 0000000000000000000000000000000000000000..01a13699210abb91e83982f883969599fa2c3233 GIT binary patch literal 24363 zcmbTdW3VV|wgh;#jk9grwr$(CZQHhO+qP{RXWN?lCMKe9&%4pl@%~g){j11Tm6>0z zwTR_}M5yU$m?4ReZm(Y<>6!89@N5k%Ai20`Ma-?7j2&o2tn{6Xg^Uetjf`ofjBQMv z%aFEh-8Scz_d> z_+iSBl`eX3+*Um;IU8d;tsEqR3HEFla$Y*$D-n=Kc?%jE8b%mTE8$p|Cl@cyE#=)$ zqM4S@m5Fb26;z#Hfh!gxTPs;FT}ydYonL~DA?br?XV}6{I!kMfjni89070jtp1^05 zRAIEhfX5jl$PX)VPOOmq6*TK)bO5v$SwROIqRru2sCMz66{oPdZh@$l2+Db6@0V8}XjgN2<()J} z;^lZ`eFZ^@Y%%BJbZW1SDT1V(EDUTNdq2B`i2t$> zRpwrb1UF_Ro0glASJ_o?3!wf5SH8hOzJ=8>1YIFSFcv^Uo<5gNi#x}JPmRW@k@4BU z)QwL?QKagx53pF-kY2t^ffVGL-loW8js6?vA&OxPg+P2EsN6ZLqbkdR0jkp${3^xV 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If $V$ is a Seifert matrix for $K$, then -$H_1(\Sigma(K), \mathbb{Z}) \cong \quot{\mathbb{Z}^n}{A\mathbb{Z}}$ where -$A = V \times V^T$, $n = \rank V$. +\[H_1(\Sigma(K), \mathbb{Z}) \cong \quot{\mathbb{Z}^n}{A\mathbb{Z}}\ \ ,\] where +$A = V \times V^T$ and $n = \rank V$. +\end{fact} %\input{ink_diag} \begin{figure}[h] \fontsize{20}{10}\selectfont diff --git a/lec_18_03.tex b/lec_18_03.tex index f2873d7..24d3aa9 100644 --- a/lec_18_03.tex +++ b/lec_18_03.tex @@ -70,9 +70,10 @@ Remark: $K \sim K^{\prime} \Leftrightarrow K \# -K^{\prime}$ is slice. \end{figure} \noindent \\ -Pontryagin-Thom construction tells us that there exists a compact oriented three - manifold $\Omega \subset B^4$ such that $\partial \Omega = Y$. +Pontryagin-Thom construction tells us that there exists a compact oriented three - manifold $\Omega \subset B^4$ such that $\partial \Omega = Y$.\\ 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 and -$\alpha, \beta \in \ker (H_1(\Sigma, \mathbb{Z}) \longrightarrow H_1(\Omega, \mathbb{Z}))$. Then there are two cycles $A, B \in \Omega$ such that $\partial A = \alpha$ and $\partial B = \beta$. +\[ +\alpha, \beta \in \ker (H_1(\Sigma, \mathbb{Z}) \longrightarrow H_1(\Omega, \mathbb{Z})).\] 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 $\partial B^+ = \beta^+$. Then @@ -151,19 +152,43 @@ Let $V = \begin{pmatrix} 0 & A\\ B & C -\end{pmatrix}$ +\end{pmatrix}$. Then \begin{align*} +tV - V^T = +\begin{pmatrix} + 0 & tA\\ + tB & tC +\end{pmatrix} +- +\begin{pmatrix} + 0 & B^T\\ + A^T & C^T +\end{pmatrix} += +\begin{pmatrix} + 0 & tA - B^T\\ + tB - A^T & tC - C^T +\end{pmatrix} +\\ \det (tV - V^T) = \det (tA - B^T) - \det(tB - A^T) \end{align*} \begin{corollary} \label{cor:slice_alex} -If $K$ is a slice knot then there exists $f \in \mathbb{Z}[t^{\pm 1}]$ such that $\Delta_K(t) = f(t) \cdot f(t^{-1})$. +If $K$ is a slice knot then there exists $f \in \mathbb{Z}[t, t^{-1}]$ such that +\[\Delta_K(t) = f(t) \cdot f(t^{-1}).\] \end{corollary} \begin{example} Figure eight knot is not slice. \end{example} \begin{fact} -If $K$ is slice, then the signature $\sigma(K) \equiv 0$. +If $K$ is slice, then the signature $\sigma(K) \equiv 0$: +\[V + V^T = +\begin{pmatrix} + 0 & A + B^T\\ + B + A^T & C + C^T +\end{pmatrix} +\Rightarrow \sigma = 0 +.\] \end{fact} diff --git a/lec_20_05.pdf b/lec_20_05.pdf new file mode 100644 index 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cf6649d..020911b 100644 --- a/lec_20_05.tex +++ b/lec_20_05.tex @@ -1,3 +1,4 @@ +% I don't have this first fragent in my notes Let $M$ be compact, oriented, connected four-dimensional manifold. If ${H_1(M, \mathbb{Z}) = 0}$ then there exists a bilinear form - the intersection form on $M$: @@ -32,6 +33,7 @@ a &\mapsto (a, \_) \in H_2(M, \mathbb{Z}) \end{align*} has coker precisely $H_1(Y, \mathbb{Z})$. \\???????????????\\ +% Here my notes begin: Let $K \subset S^3$ be a knot, $X = S^3 \setminus K$ a knot complement and $\widetilde{X} \xrightarrow{\enspace \rho \enspace} X$ an infinite cyclic cover (universal abelian cover). @@ -62,7 +64,7 @@ H_1(\widetilde{X}, \mathbb{Z}[t, t^{-1}]) &\longrightarrow \quot{\mathbb{Q}}{\ma \end{align*} \end{fact} \noindent -Note that $\mathbb{Z}$ is not PID. +Note that $\mathbb{Z}[t, t^{-1}]$ is not PID. Therefore we don't have primary decomposition of this module. We can simplify this problem by replacing $\mathbb{Z}$ by $\mathbb{R}$. We lose some date by doing this transition, but we can \begin{align*} @@ -156,7 +158,7 @@ Suppose $g = (t - \xi)^{\alpha} g^{\prime}$. Then $(t - \xi)^{k - \alpha}$ goes Every sesquilinear non-degenerate pairing \begin{align*} \quot{\Lambda}{p^k} \times \quot{\Lambda}{p} -\longleftrightarrow \frac{h}{p^k} +\longrightarrow \frac{h}{p^k} \end{align*} is isomorphic either to the pairing wit $h=1$ or to the paring with $h=-1$ depending on sign of $h(\xi)$ (which is a real number). \end{theorem} @@ -171,7 +173,7 @@ Prove in the case, when $h$ has a constant sign on $S^1$. \begin{lemma} If $P$ is a symmetric polynomial such that $P(\eta)\geq 0$ for all $\eta \in S^1$, then $P$ can be written as a product $P = g \overbar{g}$ for some polynomial $g$. \end{lemma} -\begin{proof}[Sketch of proof] +\begin{proof}[Sketch of proof]: Induction over $\deg P$.\\ Let $\zeta \notin S^1$ be a root of $P$, $P \in \mathbb{R}[t, t^{-1}]$. Assume $\zeta \notin \mathbb{R}$. We know that polynomial $P$ is divisible by $(t - \zeta)$, $(t - \overbar{\zeta})$, $(t^{-1} - \zeta)$ and $(t^{-1} - \overbar{\zeta})$. @@ -219,25 +221,23 @@ If $P$ has no roots on $S^1$ then $B(z) > 0$ for all $z$, so the assumptions of \end{proof} ?????????????????\\ \begin{align*} -(\quot{\Lambda}{p_{\xi}^k} \times -\quot{\Lambda}{p_{\xi}^k}) &\longrightarrow +\quot{\Lambda}{p_{\xi}^k} \times +\quot{\Lambda}{p_{\xi}^k} &\longrightarrow \frac{\epsilon}{p_{\xi}^k}, \quad \xi \in S^1 \setminus\{\pm 1\}\\ -(\quot{\Lambda}{q_{\xi}^k} \times -\quot{\Lambda}{q_{\xi}^k}) &\longrightarrow +\quot{\Lambda}{q_{\xi}^k} \times +\quot{\Lambda}{q_{\xi}^k} &\longrightarrow \frac{1}{q_{\xi}^k}, \quad \xi \notin S^1\\ \end{align*} ??????????????????? 1 ?? epsilon?\\ -\begin{theorem}(Matumoto, Borodzik-Conway-Politarczyk) +\begin{theorem}[Matumoto, Borodzik-Conway-Politarczyk] Let $K$ be a knot, \begin{align*} -&H_1(\widetilde{X}, \Lambda) \times +H_1(\widetilde{X}, \Lambda) \times H_1(\widetilde{X}, \Lambda) -= \bigoplus_{\substack{k, \xi, \epsilon\\ \xi in S^1}} += \bigoplus_{\substack{k, \xi, \epsilon\\ \xi \in S^1}} (\quot{\Lambda}{p_{\xi}^k}, \epsilon)^{n_k, \xi, \epsilon} \oplus \bigoplus_{k, \eta} -(\quot{\Lambda}{p_{\xi}^k})^{m_k} -\end{align*} -\begin{align*} -\text{Let } \delta_{\sigma}(\xi) = \lim_{\varepsilon \rightarrow 0^{+}} +(\quot{\Lambda}{p_{\xi}^k})^{m_k} \text{ and} \\ + \delta_{\sigma}(\xi) = \lim_{\varepsilon \rightarrow 0^{+}} \sigma(e^{2\pi i \varepsilon} \xi) - \sigma(e^{-2\pi i \varepsilon} \xi),\\ \text{then } @@ -246,7 +246,15 @@ H_1(\widetilde{X}, \Lambda) + \sigma(e^{-2 \pi i \varepsilon}\xi) \end{align*} The jump at $\xi$ is equal to -$2 \sum\limits_{k_i \text{ odd}} \epsilon_i$. The peak of the signature function is equal to $\sum\limits_{k_i \text{even}} \epsilon_i$. -%$(\eta_{k, \xi_l^{+}} -\eta_{k, \xi_l^{-}}$ +$2\sum\limits_{k_i \odd} \epsilon_i$.\\ +The peak of the signature function is equal to ${\sum\limits_{k_i \even}}{\epsilon_i}$. +\\ +????????????????? +\\ +$(\eta_{k, \xi_l^{+}} -\eta_{k, \xi_l^{-}}$ +% Livingston Pacific Jurnal of M. 2012 \end{theorem} \end{proof} + + + diff --git a/lec_25_02.pdf b/lec_25_02.pdf new file mode 100644 index 0000000000000000000000000000000000000000..5210c488727e554d894824743f99b4cc2a9dd4b3 GIT binary patch literal 23695 zcmd3NQ;?`@vTU!mZQHhO+qP}HSKGF2+qT`k+O~c7MBKPD6K9_j6X)r^{P90mR%K;n zeMKTKEK0*b%K}Mqczg8<$-s(Fk8fva3CYb(Cu(8sY~n~KYHi?bB5Y!0XKX?zZDMQY zY>v-J&%nXU3+d$SXkuUk>AumesbPmK2Jbsto5pRTHqPvWhfa-*fD0HQe2aDbYlS~$ z=mMYX*U{@ud5>v_n^g?Y@3;^iB`uv3H8Hz~dIOjczbb|lPH_XH^3c+>=-NI{T+d^SSg&|=Z0{f6e>hMQ+6bi-1fM5(wq~Qj`RH)!+ z{1s!lOQZdJc!?s{;&8gua7mOb&6(c`jNmoeOWx6v#QeshMTr5X4+iU$jqaTtT0ia9 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b/lectures_on_knot_theory.tex index 57ff132..be651af 100644 --- a/lectures_on_knot_theory.tex +++ b/lectures_on_knot_theory.tex @@ -1,3 +1,4 @@ +% use XeLaTex to compile \documentclass[12pt, twoside]{article} @@ -95,6 +96,10 @@ \DeclareMathOperator{\Lk}{lk} \DeclareMathOperator{\pt}{\{pt\}} \DeclareMathOperator{\sign}{sign} +\DeclareMathOperator{\odd}{odd} +\DeclareMathOperator{\even}{even} + + \titleformat{\subsection}{% @@ -122,642 +127,45 @@ %\input{myNotes} \section{Basic definitions \hfill\DTMdate{2019-02-25}} -%\input{lec_1.tex} +\input{lec_25_02.tex} \section{Alexander polynomial \hfill\DTMdate{2019-03-04}} -%\input{lec_2.tex} +\input{lec_04_03.tex} %add Hurewicz theorem? \section{Examples of knot classes \hfill\DTMdate{2019-03-11}} -%\input{lec_3.tex} +\input{lec_11_03.tex} \section{Concordance group \hfill\DTMdate{2019-03-18}} -%\input{lec_4.tex} +\input{lec_18_03.tex} \section{Genus $g$ cobordism \hfill\DTMdate{2019-03-25}} -%\input{lec_5.tex} +\input{lec_25_03.tex} \section{\hfill\DTMdate{2019-04-08}} -\input{lec_6.tex} +\input{lec_08_04.tex} -\section{\hfill\DTMdate{2019-04-15}} -???????????????????\\ -\begin{theorem} -Suppose that $K \subset S^3$ is a slice knot (i.e. $K$ bound a disk in $B^4$). -Then if $F$ is a Seifert surface of $K$ and $V$ denotes the associated Seifet matrix, then there exists $P \in \Gl_g(\mathbb{Z})$ such that: -\\??????????????? T ???????? -\begin{align} -PVP^{-1} = -\begin{pmatrix} -0 & A\\ -B & C -\end{pmatrix}, \quad A, C, C \in M_{g \times g} (\mathbb{Z}) -\end{align} -\end{theorem} -In other words you can find rank $g$ direct summand $\mathcal{Z}$ of $H_1(F)$ \\ -????????????\\ -such that for any -$\alpha, \beta \in \mathcal{L}$ the linking number $\Lk (\alpha, \beta^+) = 0$. -\begin{definition} -An abstract Seifert matrix (i. e. -\end{definition} -Choose a basis $(b_1, ..., b_i)$ \\ -???\\ -of $H_2(Y, \mathbb{Z}$, then $A = (b_i, b_y)$ \\??\\ is a matrix of intersection form: -\begin{align*} -\quot{\mathbb{Z}^n}{A\mathbb{Z}^n} \cong H_1(Y, \mathbb{Z}). -\end{align*} -In particular $\vert \det A\vert = \# H_1(Y, \mathbb{Z})$.\\ -That means - what is happening on boundary is a measure of degeneracy. - -\begin{center} -\begin{tikzcd} -[ - column sep=tiny, - row sep=small, - ar symbol/.style =% - {draw=none,"\textstyle#1" description,sloped}, - isomorphic/.style = {ar symbol={\cong}}, -] -H_1(Y, \mathbb{Z}) & - \times \quad H_1(Y, \mathbb{Z})& - \longrightarrow & - \quot{\mathbb{Q}}{\mathbb{Z}} - \text{ - a linking form} -\\ - \quot{\mathbb{Z}^n}{A\mathbb{Z}} \ar[u,isomorphic] & - \quot{\mathbb{Z}^n}{A\mathbb{Z}} \ar[u,isomorphic] &\\ -\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 -$A = V \times V^T$, $n = \rank V$. -%\input{ink_diag} -\begin{figure}[h] -\fontsize{20}{10}\selectfont -\centering{ -\def\svgwidth{\linewidth} -\resizebox{0.5\textwidth}{!}{\input{images/ball_4.pdf_tex}} -\caption{Pushing the Seifert surface in 4-ball.} -\label{fig:pushSeifert} -} -\end{figure} -\noindent -Let $X$ be the four-manifold obtained via the double branched cover of $B^4$ branched along $\widetilde{\Sigma}$. -\begin{fact} -\begin{itemize} -\item $X$ is a smooth four-manifold, -\item $H_1(X, \mathbb{Z}) =0$, -\item $H_2(X, \mathbb{Z}) \cong \mathbb{Z}^n$ -\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{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. -\end{align*} -???????????????????????????? -\\ -We have a primary decomposition of $H_1(Y, \mathbb{Z}) = U$ (as a group). For any $p \in \mathbb{P}$ we define $U_p$ to be the subgroup of elements annihilated by the same power of $p$. We have $U = \bigoplus_p U_p$. -\begin{example} -\begin{align*} -\text{If } U &= -\mathbb{Z}_3 \oplus -\mathbb{Z}_{45} \oplus -\mathbb{Z}_{15} \oplus -\mathbb{Z}_{75} -\text{ then }\\ -U_3 &= -\mathbb{Z}_3 \oplus -\mathbb{Z}_9 \oplus -\mathbb{Z}_3 \oplus -\mathbb{Z}_3 -\text{ and }\\ -U_5 &= -(e) \oplus -\mathbb{Z}_5 \oplus -\mathbb{Z}_5 \oplus -\mathbb{Z}_{25}. -\end{align*} -\end{example} - -\begin{lemma} -Suppose $x \in U_{p_1}$, $y \in U_{p_2}$ and $p_1 \neq p_2$. Then $ = 0$. -\end{lemma} -\begin{proof} -\begin{align*} -x \in U_{p_1} -\end{align*} -\end{proof} -\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 -bilinear form - the intersection form on $M$: - -\begin{center} -\begin{tikzcd} -[ -column sep=tiny, -row sep=small, -ar symbol/.style = {draw=none,"\textstyle#1" description,sloped}, -isomorphic/.style = {ar symbol={\cong}}, -] -H_2(M, \mathbb{Z})& -\times & H_2(M, \mathbb{Z}) -\longrightarrow & -\mathbb{Z} -\\ -\ar[u,isomorphic] \mathbb{Z}^n && &\\ -\end{tikzcd} -\end{center} -\noindent -Let us consider a specific case: $M$ has a boundary $Y = \partial M$. -Betti number $b_1(Y) = 0$, $H_1(Y, \mathbb{Z})$ is finite. -Then the intersection form can be degenerated in the sense that: - \begin{align*} -H_2(M, \mathbb{Z}) -\times H_2(M, \mathbb{Z}) -&\longrightarrow -\mathbb{Z} \quad& -H_2(M, \mathbb{Z}) &\longrightarrow \Hom (H_2(M, \mathbb{Z}), \mathbb{Z})\\ -(a, b) &\mapsto \mathbb{Z} \quad& -a &\mapsto (a, \_) H_2(M, \mathbb{Z}) -\end{align*} -has coker precisely $H_1(Y, \mathbb{Z})$. -\\???????????????\\ -Let $K \subset S^3$ be a knot, $X = S^3 \setminus K$ a knot complement and -$\widetilde{X} \xrightarrow{\enspace \rho \enspace} X$ an infinite cyclic cover (universal abelian cover). By Hurewicz theorem we know that: -\begin{align*} -\pi_1(X) \longrightarrow \quot{\pi_1(X)}{[\pi_1(X), \pi_1(X)]} = H_1(X, \mathbb{Z} ) \cong \mathbb{Z} -\end{align*} -????????????????????????????????????????????????????????????????????????\\ -????????????????????????????????????????????????????????????????????????\\ -????????????????????????????????????????????????????????????????????????\\ -????????????????????????????????????????????????????????????????????????\\ -$C_{*}(\widetilde{X})$ has a structure of a $\mathbb{Z}[t, t^{-1}] \cong \mathbb{Z}[\mathbb{Z}]$ module. \\ -Let $H_1(\widetilde{X}, \mathbb{Z}[t, t^{-1}])$ be the Alexander module of the knot $K$ with an intersection form: -\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}]} -\end{align*} - -\begin{fact} -\begin{align*} -&H_1(\widetilde{X}, \mathbb{Z}[t, t^{-1}]) \cong -\quot{\mathbb{Z}{[t, t^{-1}]}^n}{(tV - V^T)\mathbb{Z}[t, t^{-1}]^n}\;, \\ -&\text{where $V$ is a Seifert matrix.} -\end{align*} -\end{fact} -\begin{fact} -\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 -\end{align*} -\end{fact} -\noindent -Note that $\mathbb{Z}$ is not PID. -Therefore we don't have primary decomposition of this module. -We can simplify this problem by replacing $\mathbb{Z}$ by $\mathbb{R}$. We lose some date by doing this transition, but we can -\begin{align*} -\xi \in S^1 \setminus \{ \pm 1\} -&\quad -p_{\xi} = -(t - \xi)(t - \xi^{-1}) t^{-1} -\\ -\xi \in \mathbb{R} \setminus \{ \pm 1\} -&\quad -q_{\xi} = (t - \xi)(t - \xi^{-1}) t^{-1} -\\ -\xi \notin \mathbb{R} \cup S^1 -&\quad -q_{\xi} = (t - \xi)(t - \overbar{\xi})(t - \xi^{-1}) -(t - \overbar{\xi}^{-1}) t^{-2} -\end{align*} -Let $\Lambda = \mathbb{R}[t, t^{-1}]$. Then: -\begin{align*} -H_1(\widetilde{X}, \Lambda) \cong \bigoplus_{\substack{\xi \in S^1 \setminus \{\pm 1 \}\\ k\geq 0}} -( \quot{\Lambda}{p_{\xi}^k })^{n_k, \xi} -\oplus -\bigoplus_{\substack{\xi \notin S^1 \\ l\geq 0}} -(\quot{\Lambda}{q_{\xi}^l})^{n_l, \xi}& -\end{align*} -We can make this composition orthogonal with respect to the Blanchfield paring. -\vspace{0.5cm}\\ -Historical remark: -\begin{itemize} -\item John Milnor, \textit{On isometries of inner product spaces}, 1969, -\item Walter Neumann, \textit{Invariants of plane curve singularities} -%in: Knots, braids and singulari- ties (Plans-sur-Bex, 1982), 223–232, Monogr. Enseign. Math., 31, Enseignement Math., Geneva -, 1983, -\item András Némethi, \textit{The real Seifert form and the spectral pairs of isolated hypersurfaceenumerate singularities}, 1995, -%Compositio Mathematica, Volume 98 (1995) no. 1, p. 23-41 -\item Maciej Borodzik, Stefan Friedl -\textit{The unknotting number and classical invariants II}, 2014. -\end{itemize} -\vspace{0.5cm} -Let $p = p_{\xi}$, $k\geq 0$. -\begin{align*} -\quot{\Lambda}{p^k \Lambda} \times -\quot{\Lambda}{p^k \Lambda} &\longrightarrow \quot{\mathbb{Q}(t)}{\Lambda}\\ -(1, 1) &\mapsto \kappa\\ -\text{Now: } (p^k \cdot 1, 1) &\mapsto 0\\ -p^k \kappa = 0 &\in \quot{\mathbb{Q}(t)}{\Lambda}\\ -\text{therfore } p^k \kappa &\in \Lambda\\ -\text{we have } (1, 1) &\mapsto \frac{h}{p^k}\\ -\end{align*} -$h$ is not uniquely defined: $h \rightarrow h + g p^k$ doesn't affect paring. \\ -Let $h = p^k \kappa$. -\begin{example} -\begin{align*} -\phi_0 ((1, 1))=\frac{+1}{p}\\ -\phi_1 ((1, 1)) = \frac{-1}{p} -\end{align*} -$\phi_0$ and $\phi_1$ are not isomorphic. -\end{example} -\begin{proof} -Let $\Phi: -\quot{\Lambda}{p^k \Lambda} \longrightarrow - \quot{\Lambda}{p^k \Lambda}$ - be an isomorphism. \\ - Let: $\Phi(1) = g \in \lambda$ - \begin{align*} -\quot{\Lambda}{p^k \Lambda} -\xrightarrow{\enspace \Phi \enspace}& - \quot{\Lambda}{p^k \Lambda}\\ - \phi_0((1, 1)) = \frac{1}{p^k} \qquad&\qquad - \phi_1((g, g)) = \frac{1}{p^k} \quad \text{($\Phi$ is an isometry).} - \end{align*} - Suppose for the paring $\phi_1((g, g))=\frac{1}{p^k}$ we have $\phi_1((1, 1)) = \frac{-1}{p^k}$. Then: - \begin{align*} -\frac{-g\overbar{g}}{p^k} = \frac{1}{p^k} &\in \quot{\mathbb{Q}(t)}{\Lambda}\\ -\frac{-g\overbar{g}}{p^k} - \frac{1}{p^k} &\in \Lambda \\ --g\overbar{g} &\equiv 1\pmod{p} \text{ in } \Lambda\\ --g\overbar{g} - 1 &= p^k \omega \text{ for some } \omega \in \Lambda\\ -\text{evalueting at $\xi$: }\\ -\overbrace{-g(\xi)g(\xi^{-1})}^{>0} - 1 = 0 \quad \contradiction -\end{align*} -\end{proof} -????????????????????\\ -\begin{align*} -g &= \sum{g_i t^i}\\ -\overbar{g} &= \sum{g_i t^{-i}}\\ -\overbar{g}(\xi) &= \sum g_i \xi^i \quad \xi \in S^1\\ -\overbar{g}(\xi) &=\overbar{g(\xi)} -\end{align*} -Suppose $g = (t - \xi)^{\alpha} g^{\prime}$. Then $(t - \xi)^{k - \alpha}$ goes to $0$ in $\quot{\Lambda}{p^k \Lambda}$. -\begin{theorem} -Every sesquilinear non-degenerate pairing -\begin{align*} -\quot{\Lambda}{p^k} \times \quot{\Lambda}{p} -\longleftrightarrow \frac{h}{p^k} -\end{align*} -is isomorphic either to the pairing wit $h=1$ or to the paring with $h=-1$ depending on sign of $h(\xi)$ (which is a real number). -\end{theorem} -\begin{proof} -There are two steps of the proof: -\begin{enumerate} -\item -Reduce to the case when $h$ has a constant sign on $S^1$. -\item -Prove in the case, when $h$ has a constant sign on $S^1$. -\end{enumerate} -\begin{lemma} -If $P$ is a symmetric polynomial such that $P(\eta)\geq 0$ for all $\eta \in S^1$, then $P$ can be written as a product $P = g \overbar{g}$ for some polynomial $g$. -\end{lemma} -\begin{proof}[Sketch of proof] -Induction over $\deg P$.\\ -Let $\zeta \notin S^1$ be a root of $P$, $P \in \mathbb{R}[t, t^{-1}]$. Assume $\zeta \notin \mathbb{R}$. We know that polynomial $P$ is divisible by -$(t - \zeta)$, $(t - \overbar{\zeta})$, $(t^{-1} - \zeta)$ and $(t^{-1} - \overbar{\zeta})$. -Therefore: -\begin{align*} -&P^{\prime} = \frac{P}{(t - \zeta)(t - \overbar{\zeta})(t^{-1} - \zeta)(t^{-1} - \overbar{\zeta})}\\ -&P^{\prime} = g^{\prime}\overbar{g} -\end{align*} -We set $g = g^{\prime}(t - \zeta)(t - \overbar{\zeta})$ and -$P = g \overbar{g}$. Suppose $\zeta \in S^1$. Then $(t - \zeta)^2 \vert P$ (at least - otherwise it would change sign). Therefore: -\begin{align*} -&P^{\prime} = \frac{P}{(t - \zeta)^2(t^{-1} - \zeta)^2}\\ -&g = (t - \zeta)(t^{-1} - \zeta) g^{\prime} \quad \text{etc.} -\end{align*} -The map $(1, 1) \mapsto \frac{h}{p^k} = \frac{g\overbar{g}h}{p^k}$ is isometric whenever $g$ is coprime with $P$. -\end{proof} -\begin{lemma}\label{L:coprime polynomials} -Suppose $A$ and $B$ are two symmetric polynomials that are coprime and that $\forall z \in S^1$ either $A(z) > 0$ or $B(z) > 0$. Then there exist - symmetric polynomials $P$, $Q$ such that - $P(z), Q(z) > 0$ for $z \in S^1$ and $PA + QB \equiv 1$. -\end{lemma} -\begin{proof}[Idea of proof] -For any $z$ find an interval $(a_z, b_z)$ such that if $P(z) \in (a_z, b_z)$ and $P(z)A(z) + Q(z)B(z) = 1$, then $Q(z) > 0$, $x(z) = \frac{az + bz}{i}$ is a continues function on $S^1$ approximating $z$ by a polynomial . -\\??????????????????????????\\ -\begin{flalign*} -(1, 1) \mapsto \frac{h}{p^k} \mapsto \frac{g\overbar{g}h}{p^k}&\\ -g\overbar{g} h + p^k\omega = 1& -\end{flalign*} -Apply Lemma \ref{L:coprime polynomials} for $A=h$, $B=p^{2k}$. Then, if the assumptions are satisfied, -\begin{align*} -Ph + Qp^{2k} = 1\\ -p>0 \Rightarrow p = g \overbar{g}\\ -p = (t - \xi)(t - \overbar{\xi})t^{-1}\\ -\text{so } p \geq 0 \text{ on } S^1\\ -p(t) = 0 \Leftrightarrow -t = \xi or t = \overbar{\xi}\\ -h(\xi) > 0\\ -h(\overbar{\xi})>0\\ -g\overbar{g}h + Qp^{2k} = 1\\ -g\overbar{g}h \equiv 1 \mod{p^{2k}}\\ -g\overbar{g} \equiv 1 \mod{p^k} -\end{align*} -???????????????????????????????\\ -If $P$ has no roots on $S^1$ then $B(z) > 0$ for all $z$, so the assumptions of Lemma \ref{L:coprime polynomials} are satisfied no matter what $A$ is. -\end{proof} -?????????????????\\ -\begin{align*} -(\quot{\Lambda}{p_{\xi}^k} \times -\quot{\Lambda}{p_{\xi}^k}) &\longrightarrow -\frac{\epsilon}{p_{\xi}^k}, \quad \xi \in S^1 \setminus\{\pm 1\}\\ -(\quot{\Lambda}{q_{\xi}^k} \times -\quot{\Lambda}{q_{\xi}^k}) &\longrightarrow -\frac{1}{q_{\xi}^k}, \quad \xi \notin S^1\\ -\end{align*} -??????????????????? 1 ?? epsilon?\\ -\begin{theorem}(Matumoto, Borodzik-Conway-Politarczyk) -Let $K$ be a knot, -\begin{align*} -&H_1(\widetilde{X}, \Lambda) \times -H_1(\widetilde{X}, \Lambda) -= \bigoplus_{\substack{k, \xi, \epsilon\\ \xi in S^1}} -(\quot{\Lambda}{p_{\xi}^k}, \epsilon)^{n_k, \xi, \epsilon} \oplus \bigoplus_{k, \eta} -(\quot{\Lambda}{p_{\xi}^k})^{m_k} -\end{align*} -\begin{align*} -\text{Let } \delta_{\sigma}(\xi) = \lim_{\varepsilon \rightarrow 0^{+}} -\sigma(e^{2\pi i \varepsilon} \xi) -- \sigma(e^{-2\pi i \varepsilon} \xi),\\ -\text{then } -\sigma_j(\xi) = \sigma(\xi) - \frac{1}{2} \lim_{\varepsilon \rightarrow 0} -\sigma(e^{2\pi i \varepsilon}\xi) -+ \sigma(e^{-2 \pi i \varepsilon}\xi) -\end{align*} -The jump at $\xi$ is equal to -$2 \sum\limits_{k_i \text{ odd}} \epsilon_i$. The peak of the signature function is equal to $\sum\limits_{k_i \text{even}} \epsilon_i$. -%$(\eta_{k, \xi_l^{+}} -\eta_{k, \xi_l^{-}}$ -\end{theorem} -\end{proof} -\section{\hfill\DTMdate{2019-05-27}} - -.... -\begin{definition} -A square hermitian matrix $A$ of size $n$ with coefficients in \\ -the Blanchfield pairing if: -$H_1(\bar{X}$ -\end{definition} - -field of fractions - -\section{Surgery \hfill\DTMdate{2019-06-03}} -\begin{theorem} -Let $K$ be a knot and $u(K)$ its unknotting number. Let $g_4$ be a minimal four genus of a smooth surface $S$ in $B^4$ such that $\partial S = K$. Then: -\[ -u(K) \geq g_4(K) -\] -\begin{proof} -Recall that if $u(K)=u$ then $K$ bounds a disk $\Delta$ with $u$ ordinary double points. -\\ -\noindent -Remove from $\Delta$ the two self intersecting disks and glue the Seifert surface for the Hopf link. The reality surface $S$ has Euler characteristic $\chi(S) = 1 - 2u$. Therefore $g_4(S) = u$. -\end{proof} -%Tim D. Cochran and Peter Teichner -\begin{example} -The knot $8_{20}$ is slice: $\sigma \equiv 0$ almost everywhere but $\sigma(e^{\frac{ 2\pi i}{6}}) = + 1$. -\end{example} -%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 $H_1(S^1 \times S^1, \mathbb{Z}) = \mathbb{Z}^2$. 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 the 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},\\ -\phi_*(\beta) &= r \alpha + s \beta, \quad r, s \in \mathbb{Z}, \\ -\phi_* &= - \begin{pmatrix} - p & q\\ - 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} -\begin{figure}[h] -\fontsize{20}{10}\selectfont -\centering{ -\def\svgwidth{\linewidth} -\resizebox{0.5\textwidth}{!}{\input{images/dehn_twist.pdf_tex}} -\caption{Dehn twist.} -\label{fig:dehn_twist} -} -\end{figure} - - - - -\section{balagan} - -\noindent -\noindent +\section{Linking form\hfill\DTMdate{2019-04-15}} +\input{lec_15_04.tex} \section{\hfill\DTMdate{2019-05-06}} +\input{lec_06_05.tex} -\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 Nakanishi index of a knot is the minimal number of generators of $H_1(\widetilde{X})$. -\end{definition} -%see Maciej page -\noindent -Remark about notation: sometimes one writes $H_1(X; \mathbb{Z}[t, t^{-1}])$ (what is also notation for twisted homology) instead of $H_1(\widetilde{X})$. -\\ -????????????????????? -\\ -\noindent -$\Sigma_?(K) \rightarrow S^3$ ?????\\ -$H_1(\Sigma_?(K), \mathbb{Z}) = h$\\ -$H \times H \longrightarrow \quot{\mathbb{Q}}{\mathbb{Z}}$\\ -...\\ +\section{\hfill\DTMdate{2019-05-20}} +\input{lec_20_05.tex} -Let now $H = H_1(\widetilde{X})$. Can we define a paring? \\ -Let $c, d \in H(\widetilde{X})$ (see Figure \ref{fig:covering_pairing}), $\Delta$ an Alexander polynomial. We know that $\Delta c = 0 \in H_1(\widetilde{X})$ (Alexander polynomial annihilates all possible elements). Let consider a surface $F$ such that $\partial F = c$. Now consider intersection points $F \cdot d$. This points can exist in any $N_k$ or $S_k$. -\[ -\frac{1}{\Delta} \sum_{j\in \mathbb{Z} t^{-j}}(F \cdot t^j d) \in \quot{\mathbb{Q}[t, t^{-1}]}{\mathbb{Z}[t, t^{-1}]} -\] -\\ -?????????????\\ -\begin{figure}[h] -\fontsize{10}{10}\selectfont -\centering{ -\def\svgwidth{\linewidth} -\resizebox{1\textwidth}{!}{\input{images/covering_pairing.pdf_tex}} -\caption{$c, d \in H_1(\widetilde{X})$.} -\label{fig:covering_pairing} -} -\end{figure} +\section{\hfill\DTMdate{2019-05-27}} +\input{lec_27_05.tex} +\section{Surgery \hfill\DTMdate{2019-06-03}} +\input{lec_03_06.tex} -\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{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} +\section{Surgery\hfill\DTMdate{2019-06-03}} +\input{lec_10_06.tex} \end{document} + +