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\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 

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