All corrections for first lecture acording MB tips done

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*}

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}
+\end{lemma}
-\begin{fact}
+\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.

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}
+\end{lemma}
-\begin{fact}
+\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.  

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:
+	\caption{
-	an image of 
+		Not-knots examples:
-	a function ${S^1\longrightarrow S^3}$ 
+		an image of 
-	that isn't injective (left) and 
+		a function ${S^1\longrightarrow S^3}$ 
-	of a function
+		that is not injective (left) and 
-	that isn't smooth (right).}
+		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}
+\noindent
-Links:
+Example of simple links are shown in
-\begin{itemize}
+\autoref{fig:links}.
-\item
+
-a trivial link with $3$ components:
+\begin{figure}[h]
-\includegraphics[width=0.2\textwidth]{3unknots.png},
+	\centering
-\item
+	\begin{subfigure}{0.5\textwidth}
-a Hopf link: \includegraphics[width=0.13\textwidth]{Hopf.png},
+		\centering
-\item
+		\includegraphics[width=1\textwidth]
-a Whitehead link:
+		{3unknots.png}
-\includegraphics[width=0.13\textwidth]{WhiteheadLink.png},
+		\caption{A trivial link with $3$ components.}
-\item
+	\end{subfigure}
-a Borromean link:
+	\begin{subfigure}{0.4\textwidth}
-\includegraphics[width=0.1\textwidth]{BorromeanRings.png}.
+		\centering
-\end{itemize}
+		\includegraphics[width=0.7\textwidth]
-\end{example}
+		{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},
+\begin{figure}[H]	
-\includegraphics[width=0.03\textwidth]{LinkDiagram2.png},
+	\centering
-\includegraphics[width=0.05\textwidth]{LinkDiagram3.png}.
+	\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 

lec_mess.tex

@@ -2,8 +2,8 @@
 
 
 
-\begin{fact}[Milnor Singular Points of Complex Hypersurfaces]
+\begin{lemma}[Milnor Singular Points of Complex Hypersurfaces]
-\end{fact}
+\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 $K$: $K^r$. \\

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}}}