125 lines
4.5 KiB
TeX
125 lines
4.5 KiB
TeX
\begin{definition}
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Let $X$ be a knot complement.
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Then $H_1(X, \mathbb{Z}) \cong \mathbb{Z}$ and there exists an epimorphism
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$\pi_1(X) \overset{\phi}\twoheadrightarrow \mathbb{Z}$.\\
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The infinite cyclic cover of a knot complement $X$ is the cover associated with the epimorphism $\phi$.
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\[
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\widetilde{X} \longtwoheadrightarrow X
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\]
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\end{definition}
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%Rolfsen, bachalor thesis of Kamila
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\begin{figure}[h]
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\fontsize{10}{10}\selectfont
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\centering{
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\def\svgwidth{\linewidth}
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\resizebox{1\textwidth}{!}{\input{images/covering.pdf_tex}}
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\caption{Infinite cyclic cover of a knot complement.}
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\label{fig:covering}
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}
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\end{figure}
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\noindent
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\subsection{Double branched cover.}
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Let $K \subset S^3$ be a knot and $\Sigma$
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its Seifert surface.
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Let us consider a knot complement $S^3 \setminus N(K)$.
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\begin{figure}[h]
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\fontsize{10}{10}\selectfont
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\centering{
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\def\svgwidth{\linewidth}
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\resizebox{0.8\textwidth}{!}{\input{images/knot_complement.pdf_tex}}
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\caption{The double cover of the $3$-sphere branched over a knot $K$.}
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\label{fig:complement}
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}
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\end{figure}
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\noindent
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Formal sums $\sum \phi_i(t) a_i + \sum \phi_j(t)\alpha_j$ \\
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finitely generated as a $\mathbb{Z}[t, t^{-1}]$ module.
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\\
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Let $v_{ij} = \Lk(a_i, a_j^+)$. Then
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$V = \{ v_ij\}_{i, j = 1}^n$ is the Seifert matrix associated to the surface $\Sigma$ and the basis $a_1, \dots, a_n$. Therefore $a_k^+ = \sum_{j} v_{jk} \alpha_j$. Then
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$\Lk(a_i, a_k^+)= \Lk(a_k^+, a_i) = \sum_j v_{jk} \Lk(\alpha_j, a_i) = v_{ik}$.
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We also notice that $\Lk(a_i, a_j^-) = \Lk(a_i^+, a_j)= v_{ij}$ and
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$a_j^- = \sum_k v_{kj} t^{-1} \alpha_j$.
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\\
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\noindent
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The homology of $\widetilde{X}$ is generated by $a_1, \dots, a_n$ and relations.
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Let now $H = H_1(\widetilde{X})$. Can we define a paring? \\
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Let $c, d \in H(\widetilde{X})$ (see Figure \ref{fig:covering_pairing}), $\Delta$ an Alexander polynomial. We know that $\Delta c = 0 \in H_1(\widetilde{X})$ (Alexander polynomial annihilates all possible elements). Let consider a surface $F$ such that $\partial F = c$. Now consider intersection points $F \cdot d$. This points can exist in any $N_k$ or $S_k$.
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\[
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\frac{1}{\Delta} \sum_{j\in \mathbb{Z} t^{-j}}(F \cdot t^j d) \in \quot{\mathbb{Q}[t, t^{-1}]}{\mathbb{Z}[t, t^{-1}]}
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\]
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\\
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?????????????\\
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There is at least one paper where the structure of (Alexander module?) is calculated from a specific knot (?minimal number of generators?)
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\\
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C. Kearton, S. M. J. Wilson
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\\
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\begin{fact}
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Let $A$ be a matrix over principal ideal domain $R$. Than there exist matrices $C$, $D$ and $E$ such that $A = CDE$,
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\[D = \begin{bmatrix}
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d_1 & 0 & \cdots & \cdots & 0 \\
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0 & d_2 & 0 & \cdots & 0 \\
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\sdots & & \ddots & & \sdots & \\
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0 & \cdots & 0 & d_{n-1} & 0\\
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0 & \cdots & \cdots & 0 & d_n
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\end{bmatrix},\]
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where $d_{i + 1} | d_i$, and matrices
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$C$ and $E$ are invertible over $R$.\\
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$D$ is called a Smith normal form of the matrix $A$.
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\end{fact}
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\begin{definition}
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The $\mathbb{Z}[t, t^{-1}]$ module $H_1(\widetilde{X})$ is called the Alexander module of a knot $K$.
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\end{definition}
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\noindent
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Let $R$ be a PID, $M$ a finitely generated $R$ module. Let us consider
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\[
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R^k \overset{A} \longrightarrow R^n \longtwoheadrightarrow M,
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\]
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where $A$ is a $k \times n$ matrix, assume $k\ge n$. The order of $M$ is the $\gcd$ of all determinants of the $n \times n$ minors of $A$. If $k = n$ then $\ord M = \det A$.
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\begin{theorem}
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Order of $M$ doesn't depend on $A$.
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\end{theorem}
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\noindent
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For knots the order of the Alexander module is the Alexander polynomial.
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\begin{theorem}
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\[
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\forall x \in M: (\ord M) x = 0.
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\]
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\end{theorem}
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\noindent
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$M$ is well defined up to a unit in $R$.
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\\
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??????????????????\\
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General picture : $K$, $X$ knot complement...
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\begin{eqnarray*}
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H_1( X, \mathbb{Z}) = \mathbb{Z} \\
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H_1(\widetilde{X}, \mathbb{Z}[t, t^{-1}]) \\
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\pi_1(X)
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\end{eqnarray*}
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\begin{definition}
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The Nakanishi index of a knot is the minimal number of generators of $H_1(\widetilde{X})$.
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\end{definition}
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%see Maciej page
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\noindent
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Remark about notation: sometimes one writes $H_1(X; \mathbb{Z}[t, t^{-1}])$ (what is also notation for twisted homology) instead of $H_1(\widetilde{X})$.
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\\
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?????????????????????
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\\
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\noindent
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$\Sigma_?(K) \rightarrow S^3$ ?????\\
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$H_1(\Sigma_?(K), \mathbb{Z}) = h$\\
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$H \times H \longrightarrow \quot{\mathbb{Q}}{\mathbb{Z}}$\\
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...\\
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\begin{figure}[h]
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\fontsize{10}{10}\selectfont
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\centering{
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\def\svgwidth{\linewidth}
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\resizebox{1\textwidth}{!}{\input{images/covering_pairing.pdf_tex}}
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\caption{$c, d \in H_1(\widetilde{X})$.}
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\label{fig:covering_pairing}
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}
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\end{figure}
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\subsection*{Blanchfield pairing}
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