The unknotting number $u(K)$ is a number of crossing changes needed to turn a knot into an unknot, where the minimum is taken over all diagrams of a given knot.
\begin{definition}
A Gordian distance $G(K, K^\prime)$ between knots $K$ and $K^\prime$ is the minimal number of crossing changes required to turn $K$ into $K^\prime$.
\end{definition}
\begin{problem}
Prove that:
\[
G(K, K^{\prime\prime})
\leq
G(K, K^{\prime})
+
G(K^\prime, K^{\prime\prime}).
\]
Open problem:
\[
u(K\# K^\prime) = u(K) + u(K^\prime).
\]
\end{problem}
\begin{lemma}[Scharlemann '84]
Unknotting number one knots are prime.
\end{lemma}
\subsection*{Tools to bound unknotting number}
\begin{theorem}
For any symmetric polynomial $\Delta\in\mathbb{Z}[t, t^{-1}]$ such that $\Delta(1)=1$, there exists a knot $K$ such that:
\begin{enumerate}
\item
$K$ has unknotting number $1$,
\item
$\Delta_K =\Delta$.
\end{enumerate}
\end{theorem}
Let us consider a knot $K$ and its Seifert surface $\Sigma$.
the Seifert form for $K_-$
\\
the Seifert form for $K_+$
\\
$S_-+ S_+$ differs from
by a term in the bottom right corner
Let $A$ be a symmetric $n \times n$ matrix over $\mathbb{R}$. Let $A_1, \dots, A_n$ be minors of $A$. \\