delete unused
This commit is contained in:
Maria Marchwicka
parent
a633c3299b
commit
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|
|
@ -227,9 +227,9 @@ We smooth all the crossings, so we get a disjoint union of circles on the plane.
|
|||
\noindent
|
||||
Note: in general the obtained surface doesn't need to be connected, but by taking connected sum of all components we can easily get a connected surface (i.e. we take two disconnected components and cut a disk in each of them: $D_1$ and $D_2$; now we glue both components on the boundaries: $\partial D_1$ and $\partial D_2$.
|
||||
|
||||
\begin{figure}[H]
|
||||
\begin{figure}[h]
|
||||
\begin{center}
|
||||
\includegraphics[width=0.4\textwidth]{seifert_connect.png}
|
||||
\includegraphics[width=0.6\textwidth]{seifert_connect.png}
|
||||
\end{center}
|
||||
\caption{Connecting two surfaces.}
|
||||
\label{fig:SeifertConnect}
|
||||
|
|
@ -239,11 +239,11 @@ Note: in general the obtained surface doesn't need to be connected, but by takin
|
|||
Every link in $S^3$ bounds a surface $\Sigma$ that is compact, connected and orientable. Such a surface is called a Seifert surface.
|
||||
\end{theorem}
|
||||
%
|
||||
\begin{figure}[H]
|
||||
\fontsize{15}{10}\selectfont
|
||||
\begin{figure}[h]
|
||||
\fontsize{12}{10}\selectfont
|
||||
\centering{
|
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|
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|
||||
\resizebox{1\textwidth}{!}{\input{images/torus_1_2_3.pdf_tex}}
|
||||
\caption{Genus of an orientable surface.}
|
||||
\label{fig:genera}
|
||||
}
|
||||
|
|
@ -277,7 +277,7 @@ Let $\nu(\beta)$ be a tubular neighbourhood of $\beta$. The linking number can
|
|||
\begin{itemize}
|
||||
\item
|
||||
Hopf link
|
||||
\begin{figure}[H]
|
||||
\begin{figure}[h]
|
||||
\fontsize{20}{10}\selectfont
|
||||
\centering{
|
||||
\def\svgwidth{\linewidth}
|
||||
|
|
@ -286,7 +286,7 @@ Hopf link
|
|||
\end{figure}
|
||||
\item
|
||||
$T(6, 2)$ link
|
||||
\begin{figure}[H]
|
||||
\begin{figure}[h]
|
||||
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|
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@ -686,12 +686,15 @@ Let $X$ be the four-manifold obtained via the double branched cover of $B^4$ bra
|
|||
\end{fact}
|
||||
\noindent
|
||||
Let $Y = \Sigma(K)$. Then:
|
||||
\begin{align*}
|
||||
&H_1(Y, \mathbb{Z}) \times H_1(Y, \mathbb{Z}) \longrightarrow \quot{\mathbb{Q}}{\mathbb{Z}}\\ &(a,b) \mapsto a A^{-1} b^{T},\qquad
|
||||
A = V + V^T\\
|
||||
&H_1(Y, \mathbb{Z}) \cong \quot{\mathbb{Z}^n}{A\mathbb{Z}}\\
|
||||
&A \longrightarrow BAC^T \quad \text{Smith normal form}
|
||||
\end{align*}
|
||||
\begin{flalign*}
|
||||
H_1(Y, \mathbb{Z}) \times H_1(Y, \mathbb{Z}) \longrightarrow \quot{\mathbb{Q}}{\mathbb{Z}}&
|
||||
\\
|
||||
(a,b) \mapsto a A^{-1} b^{T},\qquad
|
||||
A = V + V^T&
|
||||
\\
|
||||
H_1(Y, \mathbb{Z}) \cong \quot{\mathbb{Z}^n}{A\mathbb{Z}}&\\
|
||||
A \longrightarrow BAC^T \quad \text{Smith normal form}&
|
||||
\end{flalign*}
|
||||
???????????????????????\\
|
||||
In general
|
||||
|
||||
|
|
@ -762,23 +765,23 @@ H_1(\widetilde{X}, \mathbb{Z}[t, t^{-1}]) &\longrightarrow \quot{\mathbb{Q}}{\ma
|
|||
\end{fact}
|
||||
\noindent
|
||||
Note that $\mathbb{Z}$ is not PID. Therefore we don't have primer decomposition of this moduli. We can simplify this problem by replacing $\mathbb{Z}$ by $\mathbb{R}$. We lose some date by doing this transition.
|
||||
\begin{align*}
|
||||
&\xi \in S^1 \setminus \{ \pm 1\}
|
||||
\begin{flalign*}
|
||||
\xi \in S^1 \setminus \{ \pm 1\}
|
||||
\quad
|
||||
p_{\xi} =
|
||||
(t - \xi)(1 - \xi^{-1}) t^{-1}\\
|
||||
&\xi \in \mathbb{R} \setminus \{ \pm 1\}
|
||||
(t - \xi)(1 - \xi^{-1}) t^{-1}&\\
|
||||
\xi \in \mathbb{R} \setminus \{ \pm 1\}
|
||||
\quad
|
||||
q_{\xi} = (t - \xi)(1 - \xi^{-1}) t^{-1}\\
|
||||
&\xi \notin \mathbb{R} \cup S^1 \quad
|
||||
q_{\xi} = (t - \xi)(t - \overbar{\xi})(1 - \xi^{-1})(1 - \overbar{\xi}^{-1}) t^{-2}\\
|
||||
&\Lambda = \mathbb{R}[t, t^{-1}]\\
|
||||
&\text{Then: } H_1(\widetilde{X}, \Lambda) \cong \bigoplus_{\substack{\xi \in S^1 \setminus \{\pm 1 \}\\ k\geq 0}}
|
||||
q_{\xi} = (t - \xi)(1 - \xi^{-1}) t^{-1}&\\
|
||||
\xi \notin \mathbb{R} \cup S^1 \quad
|
||||
q_{\xi} = (t - \xi)(t - \overbar{\xi})(1 - \xi^{-1})(1 - \overbar{\xi}^{-1}) t^{-2}&\\
|
||||
\Lambda = \mathbb{R}[t, t^{-1}]&\\
|
||||
\text{Then: } H_1(\widetilde{X}, \Lambda) \cong \bigoplus_{\substack{\xi \in S^1 \setminus \{\pm 1 \}\\ k\geq 0}}
|
||||
( \quot{\Lambda}{p_{\xi}^k })^{n_k, \xi}
|
||||
\oplus
|
||||
\bigoplus_{\substack{\xi \notin S^1 \\ l\geq 0}}
|
||||
(\quot{\Lambda}{q_{\xi}^l})^{n_l, \xi}
|
||||
\end{align*}
|
||||
(\quot{\Lambda}{q_{\xi}^l})^{n_l, \xi}&
|
||||
\end{flalign*}
|
||||
We can make this composition orthogonal with respect to the Blanchfield paring.
|
||||
\vspace{0.5cm}\\
|
||||
Historical remark:
|
||||
|
|
@ -893,10 +896,10 @@ Suppose $A$ and $B$ are two symmetric polynomials that are coprime and that $\fo
|
|||
\begin{proof}[Idea of proof]
|
||||
For any $z$ find an interval $(a_z, b_z)$ such that if $P(z) \in (a_z, b_z)$ and $P(z)A(z) + Q(z)B(z) = 1$, then $Q(z) > 0$, $x(z) = \frac{az + bz}{i}$ is a continues function on $S^1$ approximating $z$ by a polynomial .
|
||||
\\??????????????????????????\\
|
||||
\begin{align*}
|
||||
(1, 1) \mapsto \frac{h}{p^k} \mapsto \frac{g\overbar{g}h}{p^k}\\
|
||||
g\overbar{g} h + p^k\omega = 1
|
||||
\end{align*}
|
||||
\begin{flalign*}
|
||||
(1, 1) \mapsto \frac{h}{p^k} \mapsto \frac{g\overbar{g}h}{p^k}&\\
|
||||
g\overbar{g} h + p^k\omega = 1&
|
||||
\end{flalign*}
|
||||
Apply Lemma \ref{L:coprime polynomials} for $A=h$, $B=p^{2k}$. Then, if the assumptions are satisfied,
|
||||
\begin{align*}
|
||||
Ph + Qp^{2k} = 1\\
|
||||
|
|
@ -927,7 +930,7 @@ If $P$ has no roots on $S^1$ then $B(z) > 0$ for all $z$, so the assumptions of
|
|||
\begin{theorem}(Matumoto, Conway-Borodzik-Politarczyk)
|
||||
Let $K$ be a knot,
|
||||
\begin{align*}
|
||||
H_1(\widetilde{X}, \Lambda) \times
|
||||
&H_1(\widetilde{X}, \Lambda) \times
|
||||
H_1(\widetilde{X}, \Lambda)
|
||||
= \bigoplus_{\substack{k, \xi, \epsilon\\ \xi in S^1}}
|
||||
(\quot{\Lambda}{p_{\xi}^k}, \epsilon)^{n_k, \xi, \epsilon} \oplus \bigoplus_{k, \eta}
|
||||
|
|
|
|||
Loading…
Reference in New Issue