141 lines
4.2 KiB
TeX
141 lines
4.2 KiB
TeX
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???????
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\begin{theorem}
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Such a pairing is isometric to a pairing:
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\[
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\begin{bmatrix}
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1
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\end{bmatrix}
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\times
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\begin{bmatrix}
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1
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\end{bmatrix}
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\rightarrow
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\frac{\epsilon}{p^k_{\xi}},
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\: \epsilon \in {\pm 1}
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\]
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\end{theorem}
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?????????????
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\[
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\begin{bmatrix}
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1
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\end{bmatrix} = 1 \in \quot{\Lambda}{p^k_{\xi} \Lambda }
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\]
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????????
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\begin{theorem}
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The jump of the signature function at $\xi$ is equal to
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$2 \sum\limits_{k_i \odd} \epsilon_i$. \\
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The peak of the signature function is equal to $\sum\limits_{k_i \even} \epsilon_i$.
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\[
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(\quot{\Lambda}{p^{k_1} \Lambda}, \epsilon_1) \oplus \dots \oplus (\quot{\Lambda}{p^{k_n} \Lambda}, \epsilon_n)
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\]
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%$(\eta_{k, \xi_l^{+}} -\eta_{k, \xi_l^{-}}$
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\end{theorem}
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\begin{definition}
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A matrix $A$ is called Hermitian if
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$\overline{A(t)} = {A(t)}^T$
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\end{definition}
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\begin{theorem}[Borodzik-Friedl 2015, Borodzik-Conway-Politarczyk 2018]
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A square Hermitian matrix $A(t)$ of size $n$ with coefficients in $\mathbb{Z}[t, t^{-1}]$
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(or $\mathbb{R}[t, t^{-1}]$ ) represents
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the Blanchfield pairing if:
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\begin{eqnarray*}
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H_1(\bar{X}, \Lambda) = \quot{\Lambda^n }{A\Lambda^n },\\
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(x, y) \mapsto {\overline{x}}^T A^{-1} y \in \quot{\Omega}{\Lambda}\\
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H_1(\widetilde{X}, \Lambda) \times
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H_1(\widetilde{X}, \Lambda) \longrightarrow
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\quot{\Omega}{\Lambda},
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\end{eqnarray*}
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where $\Lambda = \mathbb{Z}[t, t^{-1}]$ or $\mathbb{R}[t, t^{-1}]$, $\Omega = \mathbb{Q}(t)$ or $\mathbb{R}(t)$
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\end{theorem}
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????????\\field of fractions ??????
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\begin{eqnarray*}
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H_1(\Sigma(K), \mathbb{Z}) = \quot{\mathbb{Z}^n}{(V + V^T) \mathbb{Z}^n}\\
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H_1(\Sigma(K), \mathbb{Z})
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\times
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H_1(\Sigma(K), \mathbb{Z})
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\longrightarrow
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= \quot{\mathbb{Q}}{\mathbb{Z}}\\
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(a, b) \mapsto a{(V + V^T)}^{-1} b
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\end{eqnarray*}
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???????????????????\\
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???????????????????\\
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\begin{eqnarray*}
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y \mapsto y + Az \\
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\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} \\
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\overline{x^T} \mathbb{1} z \in \Lambda \\
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H_1(\widetilde{X}, \Lambda) =
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\quot{ \Lambda^n }{(Vt - V) \Lambda^n}
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\\
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(a, b) \mapsto \overline{a^T}(Vt - V^T)^{-1} (t -1)b
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\end{eqnarray*}
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(Blanchfield '59)
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\begin{theorem}[Kearton '75, Friedl, Powell '15]
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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.
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\end{theorem}
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Remark:
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\begin{enumerate}
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\item
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Over $\mathbb{R}$ we can take $A$ to be diagonal.
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\item
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The jump of signature function at $\xi$ is
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equal to
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\[
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\lim_{t \rightarrow 0^+} \sign A (e^{it} \xi) - \sign A(e^{-it} \xi).
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\]
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\item
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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.
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\end{enumerate}
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\subsection{The unknotting number}
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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
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\begin{align*}
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\PICorientpluscross \mapsto \PICorientminuscross ,\\
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\PICorientminuscross \mapsto\PICorientminuscross.
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\end{align*}
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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.
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\begin{definition}
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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 $K^\prime$.
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\end{definition}
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\begin{problem}
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Prove that:
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\[
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G(K, K^{\prime\prime})
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\leq
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G(K, K^{\prime})
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+
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G(K^\prime, K^{\prime\prime}).
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\]
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Open problem:
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\[
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u(K\# K^\prime) = u(K) + u(K^\prime).
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\]
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\end{problem}
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\begin{lemma}[Scharlemann '84]
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Unknotting number one knots are prime.
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\end{lemma}
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\subsection*{Tools to bound unknotting number}
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\begin{theorem}
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For any symmetric polynomial $\Delta \in \mathbb{Z}[t, t^{-1}]$ such that $\Delta(1) = 1$, there exists a knot $K$ such that:
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\begin{enumerate}
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\item
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$K$ has unknotting number $1$,
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\item
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$\Delta_K = \Delta$.
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\end{enumerate}
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\end{theorem}
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Let us consider a knot $K$ and its Seifert surface $\Sigma$.
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the Seifert form for $K_-$
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\\
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the Seifert form for $K_+$
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\\
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$S_- + S_+$ differs from
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by a term in the bottom right corner
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Let $A$ be a symmetric $n \times n$ matrix over $\mathbb{R}$. Let $A_1, \dots, A_n$ be minors of $A$. \\
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Let $\epsilon_0 = 1$
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If |