delete unused

This commit is contained in:
Maria Marchwicka 2019-06-07 14:44:18 +02:00
parent a633c3299b
commit 0e9565df05
30 changed files with 4085 additions and 9500 deletions

719
images/BorromeanRings.svg Normal file
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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 \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 $\partial D_2$. 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} \begin{center}
\includegraphics[width=0.4\textwidth]{seifert_connect.png} \includegraphics[width=0.6\textwidth]{seifert_connect.png}
\end{center} \end{center}
\caption{Connecting two surfaces.} \caption{Connecting two surfaces.}
\label{fig:SeifertConnect} \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. 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} \end{theorem}
% %
\begin{figure}[H] \begin{figure}[h]
\fontsize{15}{10}\selectfont \fontsize{12}{10}\selectfont
\centering{ \centering{
\def\svgwidth{\linewidth} \def\svgwidth{\linewidth}
\resizebox{0.8\textwidth}{!}{\input{images/torus_1_2_3.pdf_tex}} \resizebox{1\textwidth}{!}{\input{images/torus_1_2_3.pdf_tex}}
\caption{Genus of an orientable surface.} \caption{Genus of an orientable surface.}
\label{fig:genera} \label{fig:genera}
} }
@ -277,7 +277,7 @@ Let $\nu(\beta)$ be a tubular neighbourhood of $\beta$. The linking number can
\begin{itemize} \begin{itemize}
\item \item
Hopf link Hopf link
\begin{figure}[H] \begin{figure}[h]
\fontsize{20}{10}\selectfont \fontsize{20}{10}\selectfont
\centering{ \centering{
\def\svgwidth{\linewidth} \def\svgwidth{\linewidth}
@ -286,7 +286,7 @@ Hopf link
\end{figure} \end{figure}
\item \item
$T(6, 2)$ link $T(6, 2)$ link
\begin{figure}[H] \begin{figure}[h]
\fontsize{20}{10}\selectfont \fontsize{20}{10}\selectfont
\centering{ \centering{
\def\svgwidth{\linewidth} \def\svgwidth{\linewidth}
@ -686,12 +686,15 @@ Let $X$ be the four-manifold obtained via the double branched cover of $B^4$ bra
\end{fact} \end{fact}
\noindent \noindent
Let $Y = \Sigma(K)$. Then: Let $Y = \Sigma(K)$. Then:
\begin{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 H_1(Y, \mathbb{Z}) \times H_1(Y, \mathbb{Z}) \longrightarrow \quot{\mathbb{Q}}{\mathbb{Z}}&
A = V + V^T\\ \\
&H_1(Y, \mathbb{Z}) \cong \quot{\mathbb{Z}^n}{A\mathbb{Z}}\\ (a,b) \mapsto a A^{-1} b^{T},\qquad
&A \longrightarrow BAC^T \quad \text{Smith normal form} A = V + V^T&
\end{align*} \\
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 In general
@ -762,23 +765,23 @@ H_1(\widetilde{X}, \mathbb{Z}[t, t^{-1}]) &\longrightarrow \quot{\mathbb{Q}}{\ma
\end{fact} \end{fact}
\noindent \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. 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*} \begin{flalign*}
&\xi \in S^1 \setminus \{ \pm 1\} \xi \in S^1 \setminus \{ \pm 1\}
\quad \quad
p_{\xi} = p_{\xi} =
(t - \xi)(1 - \xi^{-1}) t^{-1}\\ (t - \xi)(1 - \xi^{-1}) t^{-1}&\\
&\xi \in \mathbb{R} \setminus \{ \pm 1\} \xi \in \mathbb{R} \setminus \{ \pm 1\}
\quad \quad
q_{\xi} = (t - \xi)(1 - \xi^{-1}) t^{-1}\\ q_{\xi} = (t - \xi)(1 - \xi^{-1}) t^{-1}&\\
&\xi \notin \mathbb{R} \cup S^1 \quad \xi \notin \mathbb{R} \cup S^1 \quad
q_{\xi} = (t - \xi)(t - \overbar{\xi})(1 - \xi^{-1})(1 - \overbar{\xi}^{-1}) t^{-2}\\ q_{\xi} = (t - \xi)(t - \overbar{\xi})(1 - \xi^{-1})(1 - \overbar{\xi}^{-1}) t^{-2}&\\
&\Lambda = \mathbb{R}[t, t^{-1}]\\ \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}} \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} ( \quot{\Lambda}{p_{\xi}^k })^{n_k, \xi}
\oplus \oplus
\bigoplus_{\substack{\xi \notin S^1 \\ l\geq 0}} \bigoplus_{\substack{\xi \notin S^1 \\ l\geq 0}}
(\quot{\Lambda}{q_{\xi}^l})^{n_l, \xi} (\quot{\Lambda}{q_{\xi}^l})^{n_l, \xi}&
\end{align*} \end{flalign*}
We can make this composition orthogonal with respect to the Blanchfield paring. We can make this composition orthogonal with respect to the Blanchfield paring.
\vspace{0.5cm}\\ \vspace{0.5cm}\\
Historical remark: 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] \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 . 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*} \begin{flalign*}
(1, 1) \mapsto \frac{h}{p^k} \mapsto \frac{g\overbar{g}h}{p^k}\\ (1, 1) \mapsto \frac{h}{p^k} \mapsto \frac{g\overbar{g}h}{p^k}&\\
g\overbar{g} h + p^k\omega = 1 g\overbar{g} h + p^k\omega = 1&
\end{align*} \end{flalign*}
Apply Lemma \ref{L:coprime polynomials} for $A=h$, $B=p^{2k}$. Then, if the assumptions are satisfied, Apply Lemma \ref{L:coprime polynomials} for $A=h$, $B=p^{2k}$. Then, if the assumptions are satisfied,
\begin{align*} \begin{align*}
Ph + Qp^{2k} = 1\\ 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) \begin{theorem}(Matumoto, Conway-Borodzik-Politarczyk)
Let $K$ be a knot, Let $K$ be a knot,
\begin{align*} \begin{align*}
H_1(\widetilde{X}, \Lambda) \times &H_1(\widetilde{X}, \Lambda) \times
H_1(\widetilde{X}, \Lambda) H_1(\widetilde{X}, \Lambda)
= \bigoplus_{\substack{k, \xi, \epsilon\\ \xi in S^1}} = \bigoplus_{\substack{k, \xi, \epsilon\\ \xi in S^1}}
(\quot{\Lambda}{p_{\xi}^k}, \epsilon)^{n_k, \xi, \epsilon} \oplus \bigoplus_{k, \eta} (\quot{\Lambda}{p_{\xi}^k}, \epsilon)^{n_k, \xi, \epsilon} \oplus \bigoplus_{k, \eta}