diff --git a/lec_04_03.tex b/lec_04_03.tex index d2e89bf..7064a05 100644 --- a/lec_04_03.tex +++ b/lec_04_03.tex @@ -183,9 +183,9 @@ $S = \begin{pmatrix} \Rightarrow \text{trefoil is not trivial.} \] \end{example} -\begin{fact} +\begin{lemma} $\Delta_K(t)$ is symmetric. -\end{fact} +\end{lemma} \begin{proof} Let $S$ be an $n \times n$ matrix. \begin{align*} diff --git a/lec_11_03.tex b/lec_11_03.tex index 4b35f79..1fbddb7 100644 --- a/lec_11_03.tex +++ b/lec_11_03.tex @@ -152,21 +152,21 @@ A knot (link) is called alternating if it admits an alternating diagram. \begin{definition} A reducible crossing in a knot diagram is a crossing for which we can find a circle such that its intersection with a knot diagram is exactly that crossing. A knot diagram without reducible crossing is called reduced. \end{definition} -\begin{fact} +\begin{lemma} Any reduced alternating diagram has minimal number of crossings. -\end{fact} +\end{lemma} \begin{definition} The writhe of the diagram is the difference between the number of positive and negative crossings. \end{definition} -\begin{fact}[Tait] +\begin{lemma}[Tait] Any two diagrams of the same alternating knot have the same writhe. -\end{fact} -\begin{fact} +\end{lemma} +\begin{lemma} An alternating knot has Alexander polynomial of the form: $ a_1t^{n_1} + a_2t^{n_2} + \dots + a_s t^{n_s} $, where $n_1 < n_2 < \dots < n_s$ and $a_ia_{i+1} < 0$. -\end{fact} +\end{lemma} \begin{problem}[open] What is the minimal $\alpha \in \mathbb{R}$ such that if $z$ is a root of the Alexander polynomial of an alternating knot, then $\Re(z) > \alpha$.\\ Remark: alternating knots have very simple knot homologies. diff --git a/lec_20_05.tex b/lec_20_05.tex index 70cf05d..543bd13 100644 --- a/lec_20_05.tex +++ b/lec_20_05.tex @@ -49,20 +49,20 @@ 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{lemma} \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} +\end{lemma} +\begin{lemma} \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} +\end{lemma} \noindent Note that $\mathbb{Z}[t, t^{-1}]$ is not PID. Therefore we don't have primary decomposition of this module. diff --git a/lec_25_02.tex b/lec_25_02.tex index 5c7a482..b8d0b81 100644 --- a/lec_25_02.tex +++ b/lec_25_02.tex @@ -17,12 +17,12 @@ are shown respectively in \begin{figure}[h] \centering - \begin{subfigure}{0.3\textwidth} + \begin{subfigure}{0.45\textwidth} \centering \includegraphics[width=0.5\textwidth] {unknot.png} \end{subfigure} - \begin{subfigure}{0.3\textwidth} + \begin{subfigure}{0.45\textwidth} \centering \includegraphics[width=0.5\textwidth] {trefoil.png} @@ -34,22 +34,24 @@ are shown respectively in \begin{figure}[h] \centering - \begin{subfigure}{0.3\textwidth} + \begin{subfigure}{0.45\textwidth} \centering \includegraphics[width=0.5\textwidth] {not_injective_knot.png} \end{subfigure} - \begin{subfigure}{0.3\textwidth} + \begin{subfigure}{0.45\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).} + \caption{ + Not-knots examples: + an image of + a function ${S^1\longrightarrow S^3}$ + that is not injective (left) and + of a function + that is not smooth (right). + } \label{fig:notknot} \end{figure} @@ -101,23 +103,40 @@ are shown respectively in in $S^3$. \end{definition} -\begin{example} -Links: -\begin{itemize} -\item -a trivial link with $3$ components: -\includegraphics[width=0.2\textwidth]{3unknots.png}, -\item -a Hopf link: \includegraphics[width=0.13\textwidth]{Hopf.png}, -\item -a Whitehead link: -\includegraphics[width=0.13\textwidth]{WhiteheadLink.png}, -\item -a Borromean link: -\includegraphics[width=0.1\textwidth]{BorromeanRings.png}. -\end{itemize} -\end{example} -% +\noindent +Example of simple links are shown in +\autoref{fig:links}. + +\begin{figure}[h] + \centering + \begin{subfigure}{0.5\textwidth} + \centering + \includegraphics[width=1\textwidth] + {3unknots.png} + \caption{A trivial link with $3$ components.} + \end{subfigure} + \begin{subfigure}{0.4\textwidth} + \centering + \includegraphics[width=0.7\textwidth] + {Hopf.png} + \caption{A Hopf link.} + \end{subfigure} + \begin{subfigure}{0.4\textwidth} + \centering + \includegraphics[width=0.8\textwidth] + {WhiteheadLink.png}, + \caption{A Whitehead link.} + \end{subfigure} + \begin{subfigure}{0.4\textwidth} + \centering + \includegraphics[width=0.7\textwidth] + {BorromeanRings.png} + \caption{A Borromean link.} + \end{subfigure} + \caption{Link examples.} + \label{fig:links} +\end{figure} + % % \begin{definition}\label{def:link_diagram} @@ -133,12 +152,24 @@ a Borromean link: \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}. - +\begin{figure}[H] + \centering + \begin{subfigure}{0.1\textwidth} + \includegraphics[width=0.8\textwidth] + {LinkDiagram1.png}, + \end{subfigure} + \begin{subfigure}{0.1\textwidth} + \includegraphics[width=0.6\textwidth] + {LinkDiagram2.png}, + \end{subfigure} + \begin{subfigure}{0.1\textwidth} + \includegraphics[width=0.8\textwidth] + {LinkDiagram3.png}. + \end{subfigure} +\end{figure} \noindent There are under- and overcrossings (tunnels and bridges) on a link diagrams with an obvious meaning. @@ -152,7 +183,9 @@ We can distinguish two types of crossings: right-handed $\left(\PICorientpluscross\right)$, called a positive crossing, and left-handed $\left(\PICorientminuscross\right)$, called a negative crossing. \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}, @@ -179,8 +212,10 @@ A Reidemeister move is one of the three types of operation on a link diagram as %Authors: Arnold, V.I., Varchenko, Alexander, Gusein-Zade, S.M. \subsection{Seifert surface} + \noindent -Let $D$ be an oriented diagram of a link $L$. We change the diagram by smoothing each crossing: +Let $D$ be an oriented diagram of a link $L$. +We change the diagram by smoothing each crossing: \begin{align*} \PICorientpluscross \mapsto \PICorientLRsplit,\\ @@ -192,7 +227,8 @@ 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. +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 $\partial \Sigma = L$.\\ @@ -208,6 +244,7 @@ such that $\partial \Sigma = L$.\\ \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] @@ -312,13 +349,13 @@ Seifert surfaces of minimal genus \label{fig:unknot} \end{figure} -\begin{fact} +\begin{lemma} $ 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 a surface $\Sigma$. -\end{fact} +\end{lemma} \subsection{Seifert matrix} Let $L$ be a link and diff --git a/lec_mess.tex b/lec_mess.tex index 9541a40..836e4f4 100644 --- a/lec_mess.tex +++ b/lec_mess.tex @@ -2,8 +2,8 @@ -\begin{fact}[Milnor Singular Points of Complex Hypersurfaces] -\end{fact} +\begin{lemma}[Milnor Singular Points of Complex Hypersurfaces] +\end{lemma} %\end{comment} \noindent An oriented knot is called negative amphichiral if the mirror image $m(K)$ of $K$ is equivalent the reverse knot of $K$: $K^r$. \\ diff --git a/lectures_on_knot_theory.tex b/lectures_on_knot_theory.tex index 8728ca3..b07cbb2 100644 --- a/lectures_on_knot_theory.tex +++ b/lectures_on_knot_theory.tex @@ -58,14 +58,12 @@ {\newline}{}% \theoremstyle{break} -\newtheorem{lemma}{Lemma}[section] -\newtheorem{fact}{Fact}[section] -\newtheorem{corollary}{Corollary}[section] -\newtheorem{proposition}{Proposition}[section] -\newtheorem{example}{Example}[section] -\newtheorem{problem}{Problem}[section] -\newtheorem{definition}{Definition}[section] \newtheorem{theorem}{Theorem}[section] +\newtheorem{lemma}[theorem]{Lemma} +\newtheorem{corollary}[theorem]{Corollary} +\newtheorem{proposition}[theorem]{Proposition}\newtheorem{example}[theorem]{Example} +\newtheorem{problem}[theorem]{Problem} +\newtheorem{definition}[theorem]{Definition} \newcommand{\contradiction}{% \ensuremath{{\Rightarrow\mspace{-2mu}\Leftarrow}}}