

???????
\begin{theorem}
Such a pairing is isometric to a pairing:
\[
\begin{bmatrix}
1
\end{bmatrix}
\times 
\begin{bmatrix}
1
\end{bmatrix}
\rightarrow
\frac{\epsilon}{p^k_{\xi}}, 
\: \epsilon \in {\pm 1}
\]
\end{theorem}
?????????????
\[ 
\begin{bmatrix}
1
\end{bmatrix} = 1 \in \quot{\Lambda}{p^k_{\xi}  \Lambda }
\]
????????
\begin{theorem}
The jump of the signature function at $\xi$ is equal to 
$2 \sum\limits_{k_i \odd} \epsilon_i$. \\
The peak of the signature function is equal to $\sum\limits_{k_i \even} \epsilon_i$.
\[
(\quot{\Lambda}{p^{k_1} \Lambda}, \epsilon_1) \oplus \dots \oplus (\quot{\Lambda}{p^{k_n} \Lambda}, \epsilon_n)
\]
%$(\eta_{k, \xi_l^{+}} -\eta_{k, \xi_l^{-}}$
\end{theorem}

\begin{definition}
A matrix $A$ is called Hermitian if 
$\overline{A(t)} = {A(t)}^T$
\end{definition}

\begin{theorem}[Borodzik-Friedl 2015, Borodzik-Conway-Politarczyk 2018]
A square Hermitian matrix $A(t)$ of size $n$ with coefficients in $\mathbb{Z}[t, t^{-1}]$ 
(or $\mathbb{R}[t, t^{-1}]$ ) represents 
the Blanchfield pairing if:
\begin{eqnarray*}
H_1(\bar{X}, \Lambda) = \quot{\Lambda^n }{A\Lambda^n },\\
(x, y) \mapsto {\overline{x}}^T A^{-1} y \in \quot{\Omega}{\Lambda}\\
H_1(\widetilde{X}, \Lambda) \times 
H_1(\widetilde{X}, \Lambda) \longrightarrow
\quot{\Omega}{\Lambda},
\end{eqnarray*}
where $\Lambda = \mathbb{Z}[t, t^{-1}]$ or $\mathbb{R}[t, t^{-1}]$, $\Omega = \mathbb{Q}(t)$ or $\mathbb{R}(t)$
\end{theorem}
????????\\field of fractions ??????
\begin{eqnarray*}
H_1(\Sigma(K), \mathbb{Z}) = \quot{\mathbb{Z}^n}{(V + V^T) \mathbb{Z}^n}\\
H_1(\Sigma(K), \mathbb{Z}) 
\times
H_1(\Sigma(K), \mathbb{Z}) 
\longrightarrow
= \quot{\mathbb{Q}}{\mathbb{Z}}\\
(a, b) \mapsto a{(V + V^T)}^{-1} b
\end{eqnarray*}
???????????????????\\
???????????????????\\
\begin{eqnarray*}
y \mapsto y + Az \\
\overline{x^T} A^{-1}(y + Az) = \overline{x^T} A^{-1} y + \overline{x^T} \mathbb{1} z = \overline{x^T} A^{-1}y \in \quot{\Omega}{\Lambda} \\
\overline{x^T} \mathbb{1} z \in \Lambda \\
H_1(\widetilde{X}, \Lambda) =
\quot{ \Lambda^n }{(Vt - V) \Lambda^n}
\\
(a, b) \mapsto \overline{a^T}(Vt - V^T)^{-1} (t -1)b
\end{eqnarray*}
(Blanchfield '59)
\begin{theorem}[Kearton '75, Friedl, Powell '15]
There exits a matrix $A$ representing the Blanchfield paring over $\mathbb{Z}[t, t^{-1}]$. The size of $A$ is a size of Seifert form.
\end{theorem}
Remark:
\begin{enumerate}
\item
Over $\mathbb{R}$ we can take $A$ to be diagonal. 
\item
The jump of signature function at $\xi$ is
 equal to 
 \[
 \lim_{t \rightarrow 0^+} \sign A (e^{it} \xi) - \sign A(e^{-it} \xi).
 \]
 \item
 The minimal size of a matrix $A$ that presents a Blanchfield paring (over $\mathbb{Z}[t, t^{-1}]$)  for a knot $K$ is a knot invariant. 
\end{enumerate}

\subsection{The unknotting number}
Let $K$ be a knot and $D$ a knot diagram. A crossing change is a modification of a knot diagram by one of following changes
\begin{align*}
\PICorientpluscross \mapsto \PICorientminuscross ,\\ 
\PICorientminuscross \mapsto\PICorientminuscross.
\end{align*}
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 $A$. \\
Let $\epsilon_0 = 1$
If  