From 60d7f7d850e705619bf918fda3bba54d8bd56cd9 Mon Sep 17 00:00:00 2001 From: Maria Marchwicka Date: Thu, 31 Oct 2019 09:20:02 +0100 Subject: [PATCH] corrections of the first lecture after first review --- images/linking_hopf.pdf | Bin 3034 -> 3030 bytes images/linking_hopf.pdf_tex | 14 +- images/linking_hopf.svg | 32 +- images/linking_torus_6_2.pdf | Bin 3630 -> 3628 bytes images/linking_torus_6_2.pdf_tex | 14 +- images/linking_torus_6_2.svg | 30 +- images/seifert_matrix.pdf_tex | 14 +- images/trefoil.png | Bin 13044 -> 61992 bytes images/trefoil.svg | 1036 ++++++++++++++++++++++++++++++ images/unknot.png | Bin 4282 -> 15549 bytes images/unknot.svg | 119 ++++ lec_04_03.tex | 2 +- lec_25_02.tex | 495 ++++++++------ lec_mess.tex | 48 ++ lectures_on_knot_theory.pdf | Bin 777164 -> 855275 bytes lectures_on_knot_theory.tex | 10 +- 16 files changed, 1572 insertions(+), 242 deletions(-) create mode 100644 images/trefoil.svg create mode 100644 images/unknot.svg create mode 100644 lec_mess.tex 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zX`H8a?BnqT>vId^f3~5K$lT&qqT+;2N81Q)e}_|J${B$mC&!ifz%(-${GdveK;l~_zpJ7kLR_I= tp$}zy0N#oVF9(&UrqAKoa_^C&@&8_nISFs1pY#9#002ovPDHLkV1fW%M!x_6 diff --git a/images/unknot.svg b/images/unknot.svg new file mode 100644 index 0000000..502ab7d --- /dev/null +++ b/images/unknot.svg @@ -0,0 +1,119 @@ + + + + + + + + + + + + + image/svg+xml + + + + + + + + + + + + + + + diff --git a/lec_04_03.tex b/lec_04_03.tex index eeb7dda..d2e89bf 100644 --- a/lec_04_03.tex +++ b/lec_04_03.tex @@ -2,7 +2,7 @@ %\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} -\begin{proof}(Theorem \ref{theo:Seifert})\\ +\begin{proof}(\Cref{theo:Seifert})\\ Let $K \in S^3$ be a knot and $N = \nu(K)$ be its tubular neighbourhood. Because $K$ and $N$ are homotopy equivalent, we get: \begin{align*} H^1(S^3 \setminus N ) \cong H^1(S^3 \setminus K). diff --git a/lec_25_02.tex b/lec_25_02.tex index 3e1c85b..5c7a482 100644 --- a/lec_25_02.tex +++ b/lec_25_02.tex @@ -1,69 +1,104 @@ \begin{definition} -A knot $K$ in $S^3$ is a smooth (PL - smooth) embedding of a circle $S^1$ in $S^3$: -\begin{align*} -\varphi: S^1 \hookrightarrow S^3 -\end{align*} + A knot $K$ in $S^3$ is a smooth (PL - smooth) + embedding of a circle $S^1$ in $S^3$: + \[ + \varphi: S^1 \hookrightarrow S^3 + \] \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} +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{figure}[h] -\includegraphics[width=0.08\textwidth] -{unknot.png} -\caption{Knots examples: unknot (left) and trefoil (right).} -\label{fig:unknot} + \centering + \begin{subfigure}{0.3\textwidth} + \centering + \includegraphics[width=0.5\textwidth] + {unknot.png} + \end{subfigure} + \begin{subfigure}{0.3\textwidth} + \centering + \includegraphics[width=0.5\textwidth] + {trefoil.png} + \end{subfigure} + \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} + \centering + \begin{subfigure}{0.3\textwidth} + \centering + \includegraphics[width=0.5\textwidth] + {not_injective_knot.png} + \end{subfigure} + \begin{subfigure}{0.3\textwidth} + \centering + \includegraphics[width=0.5\textwidth] + {not_smooth_knot.png} + \end{subfigure} + \caption{Not-knots examples: + an image of + a function ${S^1\longrightarrow S^3}$ + that isn't injective (left) and + of a function + that isn't smooth (right).} + \label{fig:notknot} \end{figure} - - -\begin{itemize} -\item -Knots: -\includegraphics[width=0.08\textwidth]{unknot.png} (unknot), -\includegraphics[width=0.08\textwidth]{trefoil.png} (trefoil). -\item -Not knots: -\includegraphics[width=0.12\textwidth]{not_injective_knot.png} -(it is not an injection), -\includegraphics[width=0.08\textwidth]{not_smooth_knot.png} -(it is not smooth). -\end{itemize} -\end{example} \begin{definition} -%\hfill\\ -Two knots $K_0 = \varphi_0(S^1)$, $K_1 = \varphi_1(S^1)$ are equivalent if the embeddings $\varphi_0$ and $\varphi_1$ are isotopic, that is there exists a continues function -\begin{align*} -&\Phi: S^1 \times [0, 1] \hookrightarrow S^3, \\ -&\Phi(x, t) = \Phi_t(x) -\end{align*} - such that $\Phi_t$ is an embedding for any $t \in [0,1]$, $\Phi_0 = \varphi_0$ and -$\Phi_1 = \varphi_1$. + Two knots $K_0 = \varphi_0(S^1)$, + $K_1 = \varphi_1(S^1)$ + are equivalent if the embeddings + $\varphi_0$ and $\varphi_1$ are isotopic, + that is there exists a continues function + \begin{align*} + &\Phi: S^1 \times + [0, 1] \hookrightarrow S^3, \\ + &\Phi(x, t) = \Phi_t(x) + \end{align*} + such that + $\Phi_t$ is an embedding + for any $t \in [0,1]$, + $\Phi_0 = \varphi_0$ and + $\Phi_1 = \varphi_1$. \end{definition} \begin{theorem} -Two knots $K_0$ and $K_1$ are isotopic if and only if they are ambient isotopic, i.e. there exists a family of self-diffeomorphisms $\Psi = \{\psi_t: t \in [0, 1]\}$ such that: -\begin{align*} -&\psi(t) = \psi_t \text{ is continius on $t\in [0,1]$},\\ -&\psi_t: S^3 \hookrightarrow S^3,\\ -& \psi_0 = id ,\\ -& \psi_1(K_0) = K_1. -\end{align*} + Two knots $K_0$ and $K_1$ are isotopic + if and only if they are ambient isotopic, + i.e. there exists a family of self-diffeomorphisms + $\Psi = \{\psi_t: t \in [0, 1]\}$ such that: + \begin{align*} + &\psi(t) = \psi_t + \text{ is continius on + $t\in [0,1]$},\\ + &\psi_t: S^3 \hookrightarrow S^3,\\ + & \psi_0 = id ,\\ + & \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 $S^1$. + 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 $S^3$. + 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} @@ -85,20 +120,31 @@ a Borromean link: % % % -\begin{definition} -A link diagram $D_{\pi}$ is a picture over projection $\pi$ of a link $L$ in $\mathbb{R}^3$($S^3$) to $\mathbb{R}^2$ ($S^2$) such that: -\begin{enumerate}[label={(\arabic*)}] -\item -$D_{\pi |_L}$ is non degenerate: \includegraphics[width=0.05\textwidth]{LinkDiagram1.png}, -\item the double points are not degenerate: \includegraphics[width=0.03\textwidth]{LinkDiagram2.png}, -\item there are no triple point: \includegraphics[width=0.05\textwidth]{LinkDiagram3.png}. -\end{enumerate} +\begin{definition}\label{def:link_diagram} + A link diagram $D_{\pi}$ is a picture + over projection $\pi$ of a link $L$ in + $\mathbb{R}^3$($S^3$) to + $\mathbb{R}^2$ ($S^2$) such that: + \begin{enumerate}[label={(\arabic*)}] + \item + $D_{\pi |_L}$ is non degenerate, + \item + the double points are not degenerate, + \item there are no triple point. + \end{enumerate} \end{definition} +\noindent +By \Cref{def:link_diagram} the following pictures can not be a part of a diagram: +\includegraphics[width=0.05\textwidth]{LinkDiagram1.png}, +\includegraphics[width=0.03\textwidth]{LinkDiagram2.png}, +\includegraphics[width=0.05\textwidth]{LinkDiagram3.png}. + + \noindent There are under- and overcrossings (tunnels and bridges) on a link diagrams with an obvious meaning. -\begin{fact} +\begin{lemma} Every link admits a link diagram. -\end{fact} +\end{lemma} \noindent Let $D$ be a diagram of an oriented link (to each component of a link we add an arrow in the diagram). @@ -108,15 +154,17 @@ $\left(\PICorientpluscross\right)$, called a positive crossing, and left-handed \subsection{Reidemeister moves} A Reidemeister move is one of the three types of operation on a link diagram as shown below: \begin{enumerate}[label=\Roman*] -\item\hfill\\ -\includegraphics[width=0.6\textwidth]{rm1.png}, -\item\hfill\\\includegraphics[width=0.6\textwidth]{rm2.png}, -\item\hfill\\\includegraphics[width=0.4\textwidth]{rm3.png}. + \item\hfill\\ + \includegraphics[width=0.6\textwidth]{rm1.png}, + \item\hfill\\\includegraphics[width=0.6\textwidth]{rm2.png}, + \item\hfill\\\includegraphics[width=0.4\textwidth]{rm3.png}. \end{enumerate} \begin{theorem} [Reidemeister, 1927 ] -Two diagrams of the same link can be -deformed into each other by a finite sequence of Reidemeister moves (and isotopy of the plane). + Two diagrams of the same link can + be deformed into each other by a finite + sequence of Reidemeister moves + (and isotopy of the plane). \end{theorem} % % @@ -134,169 +182,244 @@ deformed into each other by a finite sequence of Reidemeister moves (and isotopy \noindent Let $D$ be an oriented diagram of a link $L$. We change the diagram by smoothing each crossing: \begin{align*} -\PICorientpluscross \mapsto \PICorientLRsplit,\\ -\PICorientminuscross \mapsto \PICorientLRsplit. + \PICorientpluscross \mapsto + \PICorientLRsplit,\\ + \PICorientminuscross \mapsto + \PICorientLRsplit. \end{align*} -We smooth all the crossings, so we get a disjoint union of circles on the plane. Each circle bounds a disks in $\mathbb{R}^3$ (we choose disks that don't intersect). For each smoothed crossing we add a twisted band: right-handed for a positive and left-handed for a negative one. We get an orientable surface $\Sigma$ such that $\partial \Sigma = L$.\\ +We smooth all the crossings, so we get +a disjoint union of circles on the plane. +Each circle bounds a disks in +$\mathbb{R}^3$ +(we choose disks that don't intersect). + For each smoothed crossing we add a twisted band: right-handed for a positive and left-handed for a negative one. +We get an orientable surface $\Sigma$ +such that $\partial \Sigma = L$.\\ \begin{figure}[h] -\fontsize{15}{10}\selectfont -\centering{ -\def\svgwidth{\linewidth} -\resizebox{0.8\textwidth}{!}{\input{images/seifert_alg.pdf_tex}} -\caption{Constructing a Seifert surface.} -\label{fig:SeifertAlg} -} + \fontsize{15}{10}\selectfont + \centering{ + \def\svgwidth{\linewidth} + \resizebox{0.8\textwidth}{!} + {\input{images/seifert_alg.pdf_tex}} + \caption{Constructing a Seifert surface.} + \label{fig:SeifertAlg} + } \end{figure} \noindent 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 $D_2$. Then we glue both components on the boundaries: $\partial D_1$ and $\partial D_2$. \begin{figure}[h] -\begin{center} -\includegraphics[width=0.6\textwidth]{seifert_connect.png} -\end{center} -\caption{Connecting two surfaces.} -\label{fig:SeifertConnect} + \centering + \includegraphics[width=0.6\textwidth] + {seifert_connect.png} + \caption{Connecting two surfaces.} + \label{fig:SeifertConnect} \end{figure} -\begin{theorem}[Seifert] -\label{theo:Seifert} -Every link in $S^3$ bounds a surface $\Sigma$ that is compact, connected and orientable. Such a surface is called a Seifert surface. +\begin{theorem}[Seifert]\label{theo:Seifert} + Every link in $S^3$ bounds a surface + $\Sigma$ that is compact, connected + and orientable. + Such a surface is called a Seifert surface. \end{theorem} -% + \begin{figure}[h] -\fontsize{12}{10}\selectfont -\centering{ -\def\svgwidth{\linewidth} -\resizebox{1\textwidth}{!}{\input{images/torus_1_2_3.pdf_tex}} -\caption{Genus of an orientable surface.} -\label{fig:genera} -} + \fontsize{12}{10}\selectfont + \centering + \def\svgwidth{\linewidth} + \resizebox{1\textwidth}{!}{ + \input{images/torus_1_2_3.pdf_tex}} + \caption{Genus of an orientable surface.} + \label{fig:genera} \end{figure} % % \begin{definition} -The three genus $g_3(K)$ ($g(K)$) of a knot $K$ is the minimal genus of a Seifert surface $\Sigma$ for $K$. + The three genus $g_3(K)$ ($g(K)$) + of a knot $K$ is the minimal genus + of a Seifert surface $\Sigma$ for $K$. \end{definition} \begin{corollary} -A knot $K$ is trivial if and only $g_3(K) = 0$. + A knot $K$ is trivial if and only + $g_3(K) = 0$. \end{corollary} \noindent -Remark: there are knots that admit non isotopic Seifert surfaces of minimal genus (András Juhász, 2008). +Remark: there are knots that admit non isotopic +Seifert surfaces of minimal genus +(András Juhász, 2008). \begin{definition} -Suppose $\alpha$ and $\beta$ are two simple closed curves in $\mathbb{R}^3$. -On a diagram $L$ consider all crossings between $\alpha$ and $\beta$. Let $N_+$ be the number of positive crossings, $N_-$ - negative. Then the linking number: $\Lk(\alpha, \beta) = \frac{1}{2}(N_+ - N_-)$. -\end{definition} -\begin{definition} -\label{def:lk_via_homo} -Let $\alpha$ and $\beta$ be two disjoint simple cross curves in $S^3$. -Let $\nu(\beta)$ be a tubular neighbourhood of $\beta$. The linking number can be interpreted via first homology group, where $\Lk(\alpha, \beta)$ is equal to evaluation of $\alpha$ as element of first homology group of the complement of $\beta$: -\[ -\alpha \in H_1(S^3 \setminus \nu(\beta), \mathbb{Z}) \cong \mathbb{Z}.\] + Suppose $\alpha$ and $\beta$ are two + simple closed curves in $\mathbb{R}^3$. + On a diagram $L$ consider all crossings + between $\alpha$ and $\beta$. + Let $N_+$ be the number + of positive crossings, + $N_-$ - negative. + Then the linking number: + $\Lk(\alpha, \beta) = + \frac{1}{2}(N_+ - N_-)$. +\end{definition} + +\begin{definition}\label{def:lk_via_homo} + Let $\alpha$ and $\beta$ be + two disjoint simple closed curves in $S^3$. + Let $\nu(\beta)$ be a tubular + neighbourhood of $\beta$. + The linking number can be interpreted + via first homology group, where + $\Lk(\alpha, \beta)$ is equal + to evaluation of $\alpha$ as element + of first homology group + of the complement of $\beta$: + \[ + \alpha \in H_1(S^3 \setminus + \nu(\beta), \mathbb{Z}) + \cong \mathbb{Z}. + \] \end{definition} -\begin{example} -\begin{itemize} -\item -A Hopf link: \begin{figure}[h] -\fontsize{20}{10}\selectfont -\centering{ -\def\svgwidth{\linewidth} -\resizebox{0.4\textwidth}{!}{\input{images/linking_hopf.pdf_tex}}, -} + \fontsize{10}{8}\selectfont + \centering + \def\svgwidth{\linewidth} + \resizebox{\textwidth}{!}{ +% \centering + \begin{subfigure}{0.3\textwidth} + \centering + \def\svgwidth{\linewidth} + \resizebox{1\textwidth}{!}{ + \input{images/linking_torus_6_2.pdf_tex} + } + \end{subfigure} + \begin{subfigure}{0.3\textwidth} + \centering + \def\svgwidth{\linewidth} + \resizebox{1\textwidth}{!}{ + \input{images/linking_hopf.pdf_tex} + } + \end{subfigure} + } + \vspace*{10mm} + \caption{ + Linking number of a Hopf link (left) + and a torus link $T(6, 2)$ (right). + } + \label{fig:unknot} \end{figure} -\item -$T(6, 2)$ link: -\begin{figure}[h] -\fontsize{20}{10}\selectfont -\centering{ -\def\svgwidth{\linewidth} -\resizebox{0.4\textwidth}{!}{\input{images/linking_torus_6_2.pdf_tex}}. -} -\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 $\Sigma$. +where $b_1$ is first Betti number of a surface $\Sigma$. \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 curves $\alpha_1, \dots, \alpha_n$. -Let $\alpha_1^+, \dots \alpha_n^+$ be copies of $\alpha_i$ lifted up off the surface (push up along a vector field normal to $\Sigma$). Note that elements $\alpha_i$ are contained in the Seifert surface while all $\alpha_i^+$ don't intersect the surface. -Let $\Lk(\alpha_i, \alpha_j^+) = \{a_{ij}\}$. Then the matrix $S = \{a_{ij}\}_{i, j =1}^n$ is called a Seifert matrix for $L$. Note that by choosing a different basis we get a different 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 curves +$\alpha_1, \dots, \alpha_n$. + +\noindent +Let $\alpha_1^+, \dots \alpha_n^+$ +be copies of $\alpha_i$ +lifted up off the surface +(push up along a vector field +normal to $\Sigma$). +Note that elements $\alpha_i$ are +contained in the Seifert surface while all +$\alpha_i^+$ don't intersect the surface. + +\noindent +Let $\Lk(\alpha_i, \alpha_j^+) = \{a_{ij}\}$. +Then the matrix $S = \{a_{ij}\}_{i, j =1}^n$ +is called a Seifert matrix for $L$. +Note that by choosing a different basis +we get a different matrix. \begin{figure}[h] -\fontsize{20}{10}\selectfont -\centering{ -\def\svgwidth{\linewidth} -\resizebox{0.8\textwidth}{!}{\input{images/seifert_matrix.pdf_tex}} -} + \fontsize{20}{10}\selectfont + \centering + \def\svgwidth{\linewidth} + \resizebox{0.8\textwidth}{!}{ + \input{images/seifert_matrix.pdf_tex} + } + \caption{ + A basis $\alpha_1, \alpha_2$ + of the first homology + group of a Seifert surface + and a copy of + element $\alpha_1$ pushed up + along vector normal to the Seifert surface. + } + \label{fig:alpha_plus} \end{figure} \begin{theorem} -The Seifert matrices $S_1$ and $S_2$ for the same link $L$ are S-equivalent, that is, $S_2$ can be obtained from $S_1$ by a sequence of following moves: -\begin{enumerate}[label={(\arabic*)}] - -\item -$V \rightarrow AVA^T$, where $A$ is a matrix with integer coefficients, - -\item - -$V \rightarrow -\begin{pmatrix} - \begin{array}{c|c} - V & - \begin{matrix} - \ast & 0 \\ - \sdots & \sdots\\ - \ast & 0 - \end{matrix} \\ - \hline - \begin{matrix} - \ast & \dots & \ast\\ - 0 & \dots & 0 - \end{matrix} - & - \begin{matrix} - 0 & 0\\ - 1 & 0 - \end{matrix} - \end{array} -\end{pmatrix} \quad$ -or -$\quad -V \rightarrow -\begin{pmatrix} - \begin{array}{c|c} - V & - \begin{matrix} - \ast & 0 \\ - \sdots & \sdots\\ - \ast & 0 - \end{matrix} \\ - \hline - \begin{matrix} - \ast & \dots & \ast\\ - 0 & \dots & 0 - \end{matrix} - & - \begin{matrix} - 0 & 1\\ - 0 & 0 - \end{matrix} - \end{array} -\end{pmatrix},$ -\item -inverse of (2). - -\end{enumerate} + The Seifert matrices $S_1$ and $S_2$ + for the same link $L$ are S-equivalent, + that is, $S_2$ can be obtained from + $S_1$ by a sequence of following moves: + \begin{enumerate}[label={(\arabic*)}] + \item + $V \rightarrow AVA^T$, + where $A$ is a matrix + with integer coefficients, + \item + $V \rightarrow + \begin{pmatrix} + \begin{array}{c|c} + V & + \begin{matrix} + \ast & 0 \\ + \sdots & \sdots\\ + \ast & 0 + \end{matrix} \\ + \hline + \begin{matrix} + \ast & \dots & \ast\\ + 0 & \dots & 0 + \end{matrix} + & + \begin{matrix} + 0 & 0\\ + 1 & 0 + \end{matrix} + \end{array} + \end{pmatrix} \quad$ + or + $\quad + V \rightarrow + \begin{pmatrix} + \begin{array}{c|c} + V & + \begin{matrix} + \ast & 0 \\ + \sdots & \sdots\\ + \ast & 0 + \end{matrix} \\ + \hline + \begin{matrix} + \ast & \dots & \ast\\ + 0 & \dots & 0 + \end{matrix} + & + \begin{matrix} + 0 & 1\\ + 0 & 0 + \end{matrix} + \end{array} + \end{pmatrix},$ + \item + inverse of (2). + \end{enumerate} \end{theorem} diff --git a/lec_mess.tex b/lec_mess.tex new file mode 100644 index 0000000..9541a40 --- /dev/null +++ b/lec_mess.tex @@ -0,0 +1,48 @@ + + + + +\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} diff --git a/lectures_on_knot_theory.pdf b/lectures_on_knot_theory.pdf index 040fc6624c85ae77b298274385a9eed7ebbee926..001ffc260ac31fbac493c100fafaf4bb0d7d7df0 100644 GIT binary patch delta 278742 zcmZs?WmH_zvM$=VySux)yF*BDCpa|jE{$t&2!sS^+=B#n4Fm`t+zIaP@S5y<&fE8m z_lL1ot*WnNO4aOHj5cu`e&NNUQ}9%1hY*)N{s26UE5s_((>zojPTxx}R^5ei|Os@r7l)3w*}z;Ti+e z26d6T#=#*GUB?n7h)z04cYCGxK6h6lLn3D>KNg>_##oDq&Ed>S8)mm8nEm?)`!Z)& zG=tP2Q!$A=p7XC6?Wt`)S1y`do9;wZX1w6zdPQUyMn?+75CVs%p=>h?bL|(RbF$;V z_G5V!sprTWA2y!j5adVK58TdSME0PR^!af6n`s8CPWCPdf?z^}f>79(8(6gx)p-e_5>j z0$B9P3nL5Z1UV%$oZ2T!x?jzu{17@i5{7 z$e6AEiK*=8A#R4IJXcoE)V@~ErUQA_>Kv+HR?Y`NeS!OTE1tmpFL6^A2FNCM@O@|Y z3n<}QL82eO?evCkr;2B*t1`8rMl#!5jp``L2qocRM1zzib=PK^@Szi7e;pXRiSXBk zG^8T<*mCc+BU_D^r18h8RUbfw;*pP*M!)`%J0TQ6wsB<`rTr}V1os=TO>d}5U*1F;;^?naE&IH|hxoR6`oa(n5og;0o3-%25I@KqoMDQq1fvLzHIdNDIw z&U9X178VR++#%j_(uy`F2JZlKgob$7lm|*heK(ta8eEw^vG{4K%j)b->Sld~mU6ui zVCxsCkEp0xixkIa-$iYmmzzjuqS+bWJ#;+|+$!&ErGl zAo&UEyt%^Kw^5Nvzb^5?c^3I`Z$v@(9A}1lu108q;Ky5G+Y{_EeB-4Vf0ME5tr|GL z`b?@wNMwhYM?TGH2!DPYU3$ODSrrRcI6aIr=T^0$TnA4SUdvfFNP{+8u27sJ7#Mj8 z_JmHija@E+F^t7UYpZjmcrn=)YhWW8EP@8%MiEEbeAX+`Sy`o%K!i5WjiT5xcF_we zv=tQ#EWQffrN1cpH6w;^^I(5#k=x{km2PVUzL17R>ct{t!&hK&iN ze*3w1amh30_x&uQ*ypPIokOQt)yJ8=z3!?S{zq{jlJd>sSL+7Til4jh#R4raE9LZ$ za;5CMRF^kheZ`&6v!u`^jlHTSm*8z1CMLMkzgxa*`>0G?SHs4_jj*!od!|EKyE*vz z+en_p>?mE=Xali_n)HJoQLvt9(kGNFQ1>nO_>xiKSBs0f?9_G~0vCB=vktq9lVy&Z z24;%r?I72}6ev9y%5*FtoSFC9*1pB(TO_)N7oE=y*8CY)VO~F4u$_VfrWw)H&146w zk0=#4`c@hL*ghSidmjaTIwj{=vWAXQlFM9O>mMC)yyj~80;_)V1wPJ|@tq{SH^`gW zfJgoN0aaC-|4MkTTSr-T$o4CvCmLj;wh@= zf03n7M5wa$mVFoX?_T4-`gb$gP^F;Ke$D>%4*CP7BnfP?BefAl%}9+9CT0sWY-9ie z9RXjDL0Fs=rS~qIxv8wfJ7Kpo}l4=m~q~P$NM3VTpIBW6n3_b8_J!>tD zjn%#9*tL#k2l`>onNY+k&9a&2bq6|t zOmx^Xl+jU8Igx}q+ zzDyPy8zLB^t;x=Iu;_m8;R(0U^FCUb45qnF@qF|!B}CO0l3rjp7SUz)GYnFBO#YTU zO1u3RUn8@fPt%d+XQ`|{Odq!#2s;oFBuH_FmL)%}wf|>HsX8h5*UQy?-|qbUU3mZU zYtdhKb0(av<=6Bdr9houmC|MeQ&VA2??J)vQ>2~`*G`p1cl)-<>dtF&U2(C$*?Sv2 zSj!Fc4Rork4WkJ%8D~!S+=2A*9%#CAt;ji9m%^gd)ZCjL-0(04+76D6M+Z7{Af2

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