119 lines
3.5 KiB
TeX
119 lines
3.5 KiB
TeX
\documentclass[12pt, twoside]{article}
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\usepackage{comment}
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\usepackage{amssymb}
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\usepackage{amsmath}
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\usepackage[english]{babel}
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\usepackage{csquotes}
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\usepackage{graphicx}
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\usepackage{float}
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\usepackage{titlesec}
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\usepackage{comment}
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\usepackage{pict2e}
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\usepackage{advdate}
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%... Set the first lecture date
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\ThisYear{2019}
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\ThisMonth{3}
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\ThisDay{5}
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\graphicspath{ {images/} }
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\newtheorem{lemama}{Lemma}
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\newtheorem{fact}{Fact}
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\newtheorem{example}{Example}
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%\theoremstyle{definition}
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\newtheorem{definition}{Definition}
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%\theoremstyle{plain}
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\newtheorem{theorem}{Theorem}
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\newtheorem{proposition}{Proposition}
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\input{knots_macros}
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\titleformat{\section}{\normalfont \Large \bfseries}
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{Lecture\ \thesection}{2.3ex plus .2ex}{}
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\titlespacing{\subsection}{2em}{*1}{*1}
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\begin{document}
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%\input{myNotes}
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\section{}
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\begin{definition}
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A \textbf{knot} $K$ in $S^3$ is a smooth (PL - smooth) embedding of a circle $S^1$ in $S^3$:
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\begin{align*}
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\varphi: S^1 \hookrightarrow S^3
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\end{align*}
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\end{definition}
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Usually we think about a knot as an image of an embedding: $K = \varphi(S^1)$.
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\begin{definition}
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\hfill\\
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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
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\begin{align*}
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&\Phi: S^1 \times [0, 1] \hookrightarrow S^3 \\
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&\Phi(x, t) = \Phi_t(x)
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\end{align*}
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such that $\Phi_t$ is an embedding for any $t \in [0,1]$, $\Phi_0 = \varphi_0$ and
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$\Phi_1 = \varphi_1$
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\\
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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:
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\begin{align*}
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&\Psi: S^3 \hookrightarrow S^3\\
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& \psi_0 = id\\
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& \psi_1(K_0) = K_1
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\end{align*}
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\end{definition}
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\begin{definition}
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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 $S^1$.
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\end{definition}
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\begin{definition}
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A link with k - components is a (smooth) embedding of\\ $\overbrace{S^1 \sqcup \ldots \sqcup S^1}^k$ in $S^3$
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\end{definition}
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\begin{example}
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A trivial link with $3$ components\\
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A hopf link\\
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Whitehead link\\
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Borromean link
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\end{example}
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\begin{definition}
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A link diagram is a picture over projection of a link is $S^3$/$R^3$ such that:
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\begin{enumerate}
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\item is non degenerate
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\item The double points are not degenerated
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\item There are no triple point
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\end{enumerate}
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\end{definition}
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There are under- and overcrossings (tunnels and bridges) on a link diagrams with an obvious meaning.\\
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Every link admits a link diagram.
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\subsection{Reidemeister moves}
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A Reidemeister move is one of the three types of operation on a link diagram as shown in Figure~\ref{fig: reidemeister}.
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%
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The first Reidemeister move inserts or removes a coil.
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%
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The second Reidemeister move slides a strand and inserts or removes two crossings of opposite sign.
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%
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The third Reidemeister move slides a strand over or under a crossing.
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\begin{figure}[H]
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\centering
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\includegraphics[width=0.7\textwidth]{moves.png}
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\caption{\label{fig: reidemeister}Reidemeister moves (adapted from Adams).}
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\end{figure}
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\begin{theorem} [Reidemeister’s Theorem]
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Two diagrams of the same link can be
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deformed into each other by a finite sequence of Reidemeister moves (and isotopy of the plane).
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\end{theorem}
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\section{Z nagrania Kamili}
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\begin{example}
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\begin{align*}
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&F: \mathbb{C}^2 \rightarrow \mathbb{C} \text{a polynomial} \\
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&F(0) = 0
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\end{align*}
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Fact (Milnor Singular Points of Complex Hypersurfaces):
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\end{example}
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\section{} 25.03.19
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\end{document} |