before deviding main file in many
This commit is contained in:
Maria Marchwicka
parent
20764e1b0e
commit
5d0894a257
@ -58,13 +58,13 @@
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@ -106,12 +106,12 @@
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@ -69,7 +69,8 @@
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\newcommand{\sdots}{\smash{\vdots}}
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\DeclareRobustCommand\longtwoheadrightarrow
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{\relbar\joinrel\twoheadrightarrow}
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@ -79,8 +80,11 @@
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\DeclareMathOperator{\Hom}{Hom}
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\DeclareMathOperator{\rank}{rank}
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\DeclareMathOperator{\Gl}{Gl}
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\DeclareMathOperator{\ord}{ord}
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\DeclareMathOperator{\Gl}{GL}
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\DeclareMathOperator{\Sl}{SL}
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\DeclareMathOperator{\Lk}{lk}
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\DeclareMathOperator{\pt}{\{pt\}}
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\titleformat{\section}{\normalfont \fontsize{12}{15} \bfseries}{%
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@ -562,7 +566,7 @@ If $K$ is a knot, then $n$ is necessarily even, and so $\Delta_K(t^{-1}) = t^{-n
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\begin{lemma}
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\begin{align*}
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\frac{1}{2} \deg \Delta_K(t) \leq g_3(K),
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\text{ where } deg (a_n t^n + \cdots + a_1 t^l )= k - l.
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\text{ where } deg (a_n t^n + \dots + a_1 t^l )= k - l.
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\end{align*}
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\end{lemma}
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\begin{proof}
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@ -622,12 +626,10 @@ Figure 8 knot is negative amphichiral.
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%
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\section{Concordance group \hfill\DTMdate{2019-03-18}}
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\begin{definition}
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A knot $K$ is called (smoothly) slice if $K$ is smoothly concordant to an unknot. \\
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A knot $K$ is smoothly slice if and only if $K$ bounds a smoothly embedded disk in $B^4$.
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\end{definition}
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\begin{definition}
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Two knots $K$ and $K^{\prime}$ are called (smoothly) concordant if there exists an annulus $A$ that is smoothly embedded in ${S^3 \times [0, 1]}$ such that
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${\partial A = K^{\prime} \times \{1\} \; \sqcup \; K \times \{0\}}$.
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\[
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\partial A = K^{\prime} \times \{1\} \; \sqcup \; K \times \{0\}.
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\]
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\end{definition}
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\begin{figure}[h]
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@ -637,6 +639,10 @@ ${\partial A = K^{\prime} \times \{1\} \; \sqcup \; K \times \{0\}}$.
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\resizebox{0.8\textwidth}{!}{\input{images/concordance.pdf_tex}}
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}
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\end{figure}
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\begin{definition}
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A knot $K$ is called (smoothly) slice if $K$ is smoothly concordant to an unknot. \\
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A knot $K$ is smoothly slice if and only if $K$ bounds a smoothly embedded disk in $B^4$.
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\end{definition}
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@ -681,9 +687,9 @@ Remark: $K \sim K^{\prime} \Leftrightarrow K \# -K^{\prime}$ is slice.
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\\
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\\
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\noindent
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Let $\Omega$ be an oriented \\
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Let $\Omega$ be an oriented four-manifold. \\
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???????\\
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Suppose $\Sigma$ is a Seifert matrix with an intersection form ${(\alpha, \beta) \mapsto \Lk(\alpha, \beta^+)}$. Suppose $\alpha, \beta \in H_1(\Sigma, \mathbb{Z}$ (i.e. there are cycles). \\
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Suppose $\Sigma$ is a Seifert surface and $V$ a Seifert form defined on $\Sigma$: ${(\alpha, \beta) \mapsto \Lk(\alpha, \beta^+)}$. Suppose $\alpha, \beta \in H_1(\Sigma, \mathbb{Z})$ (i.e. there are cycles). \\
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??????????????\\
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$\alpha, \beta \in \ker (H_1(\Sigma, \mathbb{Z}) \longrightarrow H_1(\Omega, \mathbb{Z}))$. Then there are two cycles $A, B \in \Omega$ such that $\partial A = \alpha$ and $\partial B = \beta$.
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Let $B^+$ be a push off of $B$ in the positive normal direction such that
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@ -692,6 +698,7 @@ Then
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$\Lk(\alpha, \beta^+) = A \cdot B^+$
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%
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%
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%ball_4_alpha_beta.pdf
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\\
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\section{\hfill\DTMdate{2019-04-08}}
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%
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@ -727,7 +734,7 @@ of $H_2(Y, \mathbb{Z}$, then $A = (b_i, b_y)$ \\??\\ is a matrix of intersection
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\begin{align*}
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\quot{\mathbb{Z}^n}{A\mathbb{Z}^n} \cong H_1(Y, \mathbb{Z}).
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\end{align*}
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In particular $\mid \det A\mid = \# H_1(Y, \mathbb{Z}$.\\
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In particular $\mid \det A\mid = \# H_1(Y, \mathbb{Z})$.\\
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That means - what is happening on boundary is a measure of degeneracy.
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\begin{center}
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@ -750,16 +757,17 @@ H_1(Y, \mathbb{Z}) &
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\end{tikzcd}
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$(a, b) \mapsto aA^{-1}b^T$
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\end{center}
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?????????????????????????????????\\
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\noindent
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The intersection form on a four-manifold determines the linking on the boundary. \\
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\noindent
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Let $K \in S^1$ be a knot, $\Sigma(K)$ its double branched cover. If $V$ is a Seifert matrix for $K$, then
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$H_1(\Sigma(K), \mathbb{Z}) \cong \quot{\mathbb{Z}^n}{A\mathbb{Z}}$ where
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$A = V \times V^T$, where $n = \rank V$.
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$A = V \times V^T$, $n = \rank V$.
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%\input{ink_diag}
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\begin{figure}[h]
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\fontsize{40}{10}\selectfont
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\fontsize{20}{10}\selectfont
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\centering{
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\def\svgwidth{\linewidth}
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\resizebox{0.5\textwidth}{!}{\input{images/ball_4.pdf_tex}}
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@ -777,20 +785,33 @@ Let $X$ be the four-manifold obtained via the double branched cover of $B^4$ bra
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\item The intersection form on $X$ is $V + V^T$.
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\end{itemize}
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\end{fact}
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\begin{figure}[h]
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\fontsize{20}{10}\selectfont
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\centering{
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\def\svgwidth{\linewidth}
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\resizebox{0.5\textwidth}{!}{\input{images/ball_4_pushed_cycle.pdf_tex}}
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\caption{Cycle pushed in 4-ball.}
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\label{fig:pushCycle}
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}
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\end{figure}
|
||||
\noindent
|
||||
Let $Y = \Sigma(K)$. Then:
|
||||
\begin{flalign*}
|
||||
H_1(Y, \mathbb{Z}) \times H_1(Y, \mathbb{Z}) \longrightarrow \quot{\mathbb{Q}}{\mathbb{Z}}&
|
||||
\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{flalign*}
|
||||
(a,b) &\mapsto a A^{-1} b^{T},\qquad
|
||||
A = V + V^T.
|
||||
\end{align*}
|
||||
????????????????????????????
|
||||
\begin{align*}
|
||||
H_1(Y, \mathbb{Z}) \cong \quot{\mathbb{Z}^n}{A\mathbb{Z}}\\
|
||||
A \longrightarrow BAC^T \quad \text{Smith normal form}
|
||||
\end{align*}
|
||||
???????????????????????\\
|
||||
In general
|
||||
|
||||
%no lecture at 29.04
|
||||
|
||||
\section{\hfill\DTMdate{2019-05-20}}
|
||||
|
||||
Let $M$ be compact, oriented, connected four-dimensional manifold. If ${H_1(M, \mathbb{Z}) = 0}$ then there exists a
|
||||
@ -1040,6 +1061,7 @@ $2 \sum\limits_{k_i \text{ odd}} \epsilon_i$. The peak of the signature function
|
||||
\end{theorem}
|
||||
\end{proof}
|
||||
\section{\hfill\DTMdate{2019-05-27}}
|
||||
|
||||
....
|
||||
\begin{definition}
|
||||
A square hermitian matrix $A$ of size $n$.
|
||||
@ -1063,8 +1085,12 @@ Remove from $\Delta$ the two self intersecting and glue the Seifert surface for
|
||||
\begin{example}
|
||||
The knot $8_{20}$ is slice: $\sigma \equiv 0$ almost everywhere but $\sigma(e^{\frac{ 2\pi i}{6}}) = + 1$.
|
||||
\end{example}
|
||||
\subsection{Surgery}
|
||||
Recall that $H_1(S^1 \times S^1, \mathbb{Z}) = \mathbb{Z}^3$. As generators for $H_1$ we can set ${\alpha = [S^1 \times \{pt\}]}$ and ${\beta=[\{pt\} \times S^1]}$. Suppose ${\phi: S^1 \times S^1 \longrightarrow S^1 \times S^1}$ is a diffeomorphism.
|
||||
%ref Structure in the classical knot concordance group
|
||||
%Tim D. Cochran, Kent E. Orr, Peter Teichner
|
||||
%Journal-ref: Comment. Math. Helv. 79 (2004) 105-123
|
||||
\subsection*{Surgery}
|
||||
%Rolfsen, geometric group theory, Diffeomorpphism of a torus, Mapping class group
|
||||
Recall that $H_1(S^1 \times S^1, \mathbb{Z}) = \mathbb{Z}^3$. As generators for $H_1$ we can set ${\alpha = [S^1 \times \pt]}$ and ${\beta=[\pt \times S^1]}$. Suppose ${\phi: S^1 \times S^1 \longrightarrow S^1 \times S^1}$ is a diffeomorphism.
|
||||
Consider an induced map on homology group:
|
||||
\begin{align*}
|
||||
H_1(S^1 \times S^1, \mathbb{Z}) \ni \phi_* (\alpha) &= p\alpha + q \beta, \quad p, q \in \mathbb{Z},\\
|
||||
@ -1075,9 +1101,30 @@ H_1(S^1 \times S^1, \mathbb{Z}) \ni \phi_* (\alpha) &= p\alpha + q \beta, \quad
|
||||
r & s
|
||||
\end{pmatrix}
|
||||
\end{align*}
|
||||
|
||||
As $\phi_*$ is diffeomorphis, it must be invertible over $\mathbb{Z}$. Then for a direction preserving diffeomorphism we have $\det \phi_* = 1$. Therefore $\phi_* \in \Sl(2, \mathbb{Z})$.
|
||||
\end{theorem}
|
||||
|
||||
\vspace{10cm}
|
||||
\begin{theorem}
|
||||
Every such a matrix can be realized as a torus.
|
||||
\end{theorem}
|
||||
\begin{proof}
|
||||
\begin{enumerate}[label={(\Roman*)}]
|
||||
\item
|
||||
Geometric reason
|
||||
\begin{align*}
|
||||
\phi_t:
|
||||
S^1 \times S^1 &\longrightarrow S^1 \times S^1 \\
|
||||
S^1 \times \pt &\longrightarrow \pt \times S^1 \\
|
||||
\pt \times S^1 &\longrightarrow S^1 \times \pt \\
|
||||
(x, y) & \mapsto (-y, x)
|
||||
\end{align*}
|
||||
\item
|
||||
\end{enumerate}
|
||||
\end{proof}
|
||||
|
||||
|
||||
|
||||
|
||||
\section{balagan}
|
||||
|
||||
@ -1128,4 +1175,87 @@ has a subspace of dimension $g_{\Sigma}$ on which it is zero:
|
||||
* & \dots & * & * & \dots & *
|
||||
\end{pmatrix}_{2g_{\Sigma} \times 2g_{\Sigma}}
|
||||
\end{align*}
|
||||
|
||||
|
||||
\section{\hfill\DTMdate{2019-05-06}}
|
||||
|
||||
\begin{definition}
|
||||
Let $X$ be a knot complement.
|
||||
Then $H_1(X, \mathbb{Z}) \cong \mathbb{Z}$ and there exists an epimorphism
|
||||
$\pi_1(X) \overset{\phi}\twoheadrightarrow \mathbb{Z}$.\\
|
||||
The infinite cyclic cover of a knot complement $X$ is the cover associated with the epimorphism $\phi$.
|
||||
\[
|
||||
\widetilde{X} \longtwoheadrightarrow X
|
||||
\]
|
||||
\end{definition}
|
||||
%Rolfsen, bachalor thesis of Kamila
|
||||
\begin{figure}[h]
|
||||
\fontsize{10}{10}\selectfont
|
||||
\centering{
|
||||
\def\svgwidth{\linewidth}
|
||||
\resizebox{1\textwidth}{!}{\input{images/covering.pdf_tex}}
|
||||
\caption{Infinite cyclic cover of a knot complement.}
|
||||
\label{fig:covering}
|
||||
}
|
||||
\end{figure}
|
||||
\begin{figure}[h]
|
||||
\fontsize{10}{10}\selectfont
|
||||
\centering{
|
||||
\def\svgwidth{\linewidth}
|
||||
\resizebox{0.8\textwidth}{!}{\input{images/knot_complement.pdf_tex}}
|
||||
\caption{A knot complement.}
|
||||
\label{fig:complement}
|
||||
}
|
||||
\end{figure}
|
||||
\noindent
|
||||
Formal sums $\sum \phi_i(t) a_i + \sum \phi_j(t)\alpha_j$ \\
|
||||
finitely generated as a $\mathbb{Z}[t, t^{-1}]$ module.
|
||||
\\
|
||||
Let $v_{ij} = \Lk(a_i, a_j^+)$. Then
|
||||
$V = \{ v_ij\}_{i, j = 1}^n$ 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
|
||||
$\Lk(a_i, a_k^+)= \Lk(a_k^+, a_i) = \sum_j v_{jk} \Lk(\alpha_j, a_i) = v_{ik}$.
|
||||
We also notice that $\Lk(a_i, a_j^-) = \Lk(a_i^+, a_j)= v_{ij}$ and
|
||||
$a_j^- = \sum_k v_{kj} t^{-1} \alpha_j$.
|
||||
\\
|
||||
\noindent
|
||||
The homology of $\widetilde{X}$ is generated by $a_1, \dots, a_n$ and relations.
|
||||
|
||||
\begin{definition}
|
||||
The $\mathbb{Z}[t, t^{-1}]$ module $H_1(\widetilde{X})$ is called the Alexander module of knot $K$.
|
||||
\end{definition}
|
||||
\noindent
|
||||
Let $R$ be a PID, $M$ a finitely generated $R$ module. Let us consider
|
||||
\[
|
||||
R^k \overset{A} \longrightarrow R^n \longtwoheadrightarrow M,
|
||||
\]
|
||||
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 $\ord M = \det A$.
|
||||
\begin{theorem}
|
||||
Order of $M$ doesn't depend on $A$.
|
||||
\end{theorem}
|
||||
\noindent
|
||||
For knots the order of the Alexander module is the Alexander polynomial.
|
||||
\begin{theorem}
|
||||
\[
|
||||
\forall x \in M: (\ord M) x = 0.
|
||||
\]
|
||||
\end{theorem}
|
||||
\noindent
|
||||
$M$ is well defined up to a unit in $R$.
|
||||
\subsection*{Blanchfield pairing}
|
||||
\section{balagan}
|
||||
\begin{theorem}
|
||||
Let $H_p$ be a $p$ - torsion part of $H$. There exists an orthogonal decomposition of $H_p$:
|
||||
\[
|
||||
H_p = H_{p, 1} \oplus \dots \oplus H_{p, r_p}.
|
||||
\]
|
||||
$H_{p, i}$ is a cyclic module:
|
||||
\[
|
||||
H_{p, i} = \quot{\mathbb{Z}[t, t^{-1}]}{p^{k_i} \mathbb{Z} [t, t^{-1}]}
|
||||
\]
|
||||
\end{theorem}
|
||||
\noindent
|
||||
The proof is the same as over $\mathbb{Z}$.
|
||||
\noindent
|
||||
|
||||
\end{document}
|
||||
|
||||
|
||||
Loading…
Reference in New Issue
Block a user