First part of the firtst lecture

lectures_on_knot_theory.tex

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+\documentclass[12pt, twoside]{article}
+
+\usepackage{comment}
+\usepackage{amssymb}
+\usepackage{amsmath}
+\usepackage[english]{babel}
+\usepackage{csquotes}
+\usepackage{graphicx}
+\usepackage{float}
+\usepackage{titlesec}
+\usepackage{comment}
+\usepackage{pict2e}
+
+\usepackage{advdate}
+
+%... Set the first lecture date
+\ThisYear{2019}
+\ThisMonth{3}
+\ThisDay{5}
+
+
+\graphicspath{ {images/} }
+
+\newtheorem{lemama}{Lemma}
+\newtheorem{fact}{Fact}
+\newtheorem{example}{Example}
+%\theoremstyle{definition}
+\newtheorem{definition}{Definition}
+%\theoremstyle{plain}
+\newtheorem{theorem}{Theorem}
+\newtheorem{proposition}{Proposition}
+
+\input{knots_macros}
+
+
+\titleformat{\section}{\normalfont \Large \bfseries}
+{Lecture\ \thesection}{2.3ex plus .2ex}{} 
+\titlespacing{\subsection}{2em}{*1}{*1}
+
+
+\begin{document}
+%\input{myNotes}
+
+\section{}
+\begin{definition}
+A \textbf{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*}   
+\end{definition}
+Usually we think about a knot as an image of an embedding: $K = \varphi(S^1)$.
+\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 $K_0$ and $K_1$ are isotopic if and only if they are ambient isotopic, i.e. there exists a family of self-diffeomorphisms $\Phi$ such that:
+\begin{align*}
+&\Psi: S^3 \hookrightarrow S^3\\
+& \psi_0 = id\\
+& \psi_1(K_0) = K_1
+\end{align*}
+\end{definition}
+\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 $S^1$.
+\end{definition}
+\begin{definition}
+A link with k - components is a (smooth) embedding of\\ $\overbrace{S^1 \sqcup \ldots \sqcup S^1}^k$ in $S^3$
+\end{definition}
+\begin{example}
+A trivial link with $3$ components\\
+A hopf link\\
+Whitehead link\\
+Borromean link
+\end{example}
+\begin{definition}
+A link diagram is a picture over projection of a link is $S^3$/$R^3$ such that:
+\begin{enumerate}
+\item is non degenerate 
+\item The double points are not degenerated 
+\item There are no triple point
+\end{enumerate}
+\end{definition}
+There are under- and overcrossings  (tunnels and bridges) on a link diagrams with an obvious meaning.\\
+Every link admits a link diagram.
+
+\subsection{Reidemeister moves}
+A Reidemeister move is one of the three types of operation on a link diagram as shown in Figure~\ref{fig: reidemeister}.
+%
+The first Reidemeister move inserts or removes a coil. 
+%
+The second Reidemeister move slides a strand and inserts or removes two crossings of opposite sign. 
+%
+The third Reidemeister move slides a strand over or under  a crossing.
+\begin{figure}[H]
+\centering
+\includegraphics[width=0.7\textwidth]{moves.png}
+\caption{\label{fig: reidemeister}Reidemeister moves (adapted from Adams).}
+\end{figure}
+\begin{theorem} [Reidemeister’s Theorem]
+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}
+\section{Z nagrania Kamili}
+\begin{example}
+\begin{align*}
+&F: \mathbb{C}^2 \rightarrow \mathbb{C}  \text{a polynomial} \\ 
+&F(0) = 0
+\end{align*}
+Fact (Milnor Singular Points of Complex Hypersurfaces):
+\end{example}
+\section{} 25.03.19 
+
+\end{document}