lectures_on_knot_theory/lec_11_03.tex

\subsection{Algebraic knots}
\noindent
Suppose $F: \mathbb{C}^2 \rightarrow \mathbb{C}$ is a polynomial and $F(0) = 0$. Let take a small sphere $S^3$ around zero. This sphere intersect set of roots of $F$ (zero set of $F$) transversally and by the implicit function theorem the intersection is a manifold. 
The dimension of sphere is $3$ and $F^{-1}(0)$ has codimension $2$. 
So there is a subspace $L$ - compact one dimensional manifold without boundary. 
That means that $L$ is a link in $S^3$. 
\begin{figure}[h]
\fontsize{40}{10}\selectfont
\centering{
\def\svgwidth{\linewidth}
\resizebox{0.2\textwidth}{!}{\input{images/milnor_singular.pdf_tex}}
}
\caption{The intersection of a sphere $S^3$ and zero set of polynomial $F$ is a link $L$.}
\label{fig:milnor_singular}
\end{figure}
%ref: Milnor Singular Points of Complex Hypersurfaces
\begin{theorem}

$L$ is an unknot if and only if 
zero is a smooth point, i.e.
$\bigtriangledown F(0) \neq 0$ (provided $S^3$ has a sufficiently small radius).
\end{theorem}
\noindent
Remark: if $S^3$ is large it can happen that $L$ is unlink, but $F^{-1}(0) \cap B^4$ is "complicated". \\
%Kyle M. Ormsby 
\noindent
In other words: if we take sufficiently small sphere, the link is non-trivial if and only if the point $0$ is singular and the isotopy type of the link doesn't depend on the radius of the sphere. 
A link obtained is such a way is called an 
algebraic link (in older books on knot theory there is another notion of algebraic link with another meaning).
%ref:  Eisenbud, D., Neumann, W.
\begin{example}
Let $p$ and $q$ be coprime numbers such that $p<q$ and $p,q>1$. \\
Zero is an isolated singular point ($\bigtriangledown F(0) = 0$). $F$ is quasi - homogeneous polynomial, so the isotopy class of the link doesn't depend on the choice of a sphere. 
Consider $S^3 = \{ (z, w) \in \mathbb{C} : \max( \vert z \vert, \vert w \vert )\} = \varepsilon$.
The intersection
$F^{-1}(0) \cap S^3$ is a torus $T(p, q)$.
\\???????????????????
$F(z, w) = z^p - w^q$\\
.\\
$F^{-1}(0) = \{t = t^q, w = t^p\}.$ For unknot $t = \max (\vert t\vert ^p, \vert t \vert^q) = \varepsilon$.
\end{example}
as a corollary we see that $K_T^{n, }$ ???? \\
is not slice unless $m=0$. \\
$t = re^{i \Theta}, \Theta \in [0, 2\pi], r = \varepsilon^{\frac{i}{p}}$

\begin{figure}[h]
\fontsize{40}{10}\selectfont
\centering{
\def\svgwidth{\linewidth}
\resizebox{0.2\textwidth}{!}{\input{images/polynomial_and_surface.pdf_tex}}
}
\caption{Sa.}
\label{fig:polynomial_and_surface}
\end{figure}
\begin{theorem}
Suppose $L$ is an algebraic link. $L = F^{-1}(0) \cap S^3$. Let 
\begin{align*}
&\varphi : S^3 \setminus L \longrightarrow S^1 \\
&\varphi(z, w) =\frac{F(z, w)}{\vert F(z, w) \vert}\in S^1, \quad (z, w) \notin F^{-1}(0).
\end{align*}
The map $\varphi$ is a locally trivial fibration.
\end{theorem}
???????\\
$ rh D \varphi \equiv 1$
\begin{definition}
A map $\Pi : E \longrightarrow B$ is locally trivial fibration with fiber $F$ if for any $b \in B$, there is a neighbourhood $U \subset B$ such that $\Pi^{-1}(U) \cong U \times $ \\
????????????\\ $\Gamma$ ?????????????\\
FIGURES\\
!!!!!!!!!!!!!!!!!!!!!!!!!!\\
\end{definition}

\begin{theorem}
The map $j: \mathscr{C} \longrightarrow \mathbb{Z}^{\infty}$ is a surjection that maps ${K_n}$ to a linear independent set. Moreover $\mathscr{C} \cong \mathbb{Z}$
\end{theorem}
...
\\
In general $h$ is defined only up to homotopy, but this means that 
\[
h_* : H_1 (F, \mathbb{Z}) \longrightarrow H_1 (F, \mathbb{Z})
\]
is well defined \\
???????????\\ map.
\begin{theorem}
\label{thm:F_as_S}
Suppose $S$ is a Seifert matrix associated with $F$ then $h = S^{-1}S^T$.
\end{theorem}
\begin{proof}
TO WRITE REFERENCE!!!!!!!!!!!
%see Arnold Varchenko vol II
%Picard - Lefschetz formula
%Nemeth (Real Seifert forms 
\end{proof}
\noindent
Consequences: 
\begin{enumerate}[label={(\arabic*)}]
\item 
the Alexander polynomial is the characteristic polynomial of $h$:
\[
\Delta_L (t) = \det (h - t I d)
\]
In particular $\Delta_L $ is monic (i.e. the top coefficient is $\pm 1$), 
????????????????
\item 
S is invertible,
\item
$F$ minimize the genus (i.e. $F$ is minimal genus Seifert surface).
\\??????????????????\\
\end{enumerate}
%
\begin{definition}
A link $L$ is fibered if there exists a map ${\phi: S^3\setminus L \longrightarrow S^1}$ which is locally trivial fibration. 
\end{definition}
\noindent
If $L$ is fibered then Theorem \ref{thm:F_as_S} holds and all its consequences. 
\begin{problem}
If $K_1$ and $K_2$ are fibered knots, then also $K_1 \# K_2$ is fibered. 
\end{problem}
\noindent
?????????????????????\\
\begin{problem}
Prove that connected sum is well defined:\\ 
$\Delta_{K_1 \# K_2} = 
\Delta_{K_1} + \Delta_{K_2}$ and
$g_3(K_1 \# K_2) = g_3(K_1) + g_3(K_2)$.

\end{problem}
\begin{figure}[h]
\fontsize{12}{10}\selectfont
\centering{
\def\svgwidth{\linewidth}
\resizebox{1\textwidth}{!}{\input{images/satellite.pdf_tex}}}
\caption{Example for a  satellite knot: a Whitehead double of a trefoil.\\ 
The pattern knot embedded non-trivially in an unknotted solid torus $T$ (e.i. $K \not\subset S^3\subset T$) on the left and the pattern in a companion knot - trefoil - on the right.}
\label{fig:sattelite}
\end{figure}
\noindent
\subsection{Alternating knot}
\begin{definition}
A knot (link) is called alternating if it admits an alternating diagram.
\end{definition}

\begin{figure}[h]
\fontsize{12}{10}\selectfont
\centering{
\def\svgwidth{\linewidth}
\includegraphics[width=0.3\textwidth]{figure8.png}
}
\caption{Example: figure eight knot is an alternating knot.}
\label{fig:fig8}
\end{figure}

\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{lemma}
Any reduced alternating diagram has minimal number of crossings.
\end{lemma}
\begin{definition}
The writhe of the diagram is the difference between the number of positive and negative crossings.
\end{definition}
\begin{lemma}[Tait]
Any two diagrams of the same alternating knot have the same writhe.
\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{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.
\end{problem}
\begin{proposition}
If $T_{p, q}$ is a torus knot, $p < q$, then it is alternating if and only if $p=2$.
\end{proposition}