100 lines
3.4 KiB
TeX
100 lines
3.4 KiB
TeX
\begin{theorem}
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Let $K$ be a knot and $u(K)$ its unknotting number. Let $g_4$ be a minimal four genus of a smooth surface $S$ in $B^4$ such that $\partial S = K$. Then:
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\[
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u(K) \geq g_4(K)
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\]
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\begin{proof}
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Recall that if $u(K)=u$ then $K$ bounds a disk $\Delta$ with $u$ ordinary double points.
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\\
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???????????????
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\\
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\begin{eqnarray*}
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\chi (D^2) = 1 \\
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\chi (\Delta) = 1 - u\\
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\gamma = 0 \in \pi_1(B^4 \setminus S)
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\end{eqnarray*}
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??????????????
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\\
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\noindent
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Remove from $\Delta$ the two self intersecting disks and glue the Seifert surface for the Hopf link. The reality surface $S$ has Euler characteristic $\chi(S) = 1 - 2u$. Therefore $g_4(S) = u$.
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\end{proof}
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%Tim D. Cochran and Peter Teichner
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\begin{example}
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The knot $8_{20}$ is slice: $\sigma \equiv 0$ almost everywhere but $\sigma(e^{\frac{ 2\pi i}{6}}) = + 1$.
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\end{example}
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%ref Structure in the classical knot concordance group
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%Tim D. Cochran, Kent E. Orr, Peter Teichner
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%Journal-ref: Comment. Math. Helv. 79 (2004) 105-123
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\subsection*{Surgery}
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%Rolfsen, geometric group theory, Diffeomorpphism of a torus, Mapping class group
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Recall that $H_1(S^1 \times S^1, \mathbb{Z}) = \mathbb{Z}^2$. 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.
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Consider an induced map on the homology group:
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\begin{align*}
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H_1(S^1 \times S^1, \mathbb{Z}) \ni \phi_* (\alpha) &= p\alpha + q \beta, \quad p, q \in \mathbb{Z},\\
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\phi_*(\beta) &= r \alpha + s \beta, \quad r, s \in \mathbb{Z}, \\
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\phi_* &=
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\begin{pmatrix}
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p & q\\
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r & s
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\end{pmatrix}.
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\end{align*}
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As $\phi_*$ is diffeomorphis, it must be invertible over $\mathbb{Z}$. Then for a direction preserving diffeomorphism we have $\det \phi_* = 1$. Therefore $\phi_* \in \Sl(2, \mathbb{Z})$.
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\end{theorem}
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\vspace{10cm}
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\begin{theorem}
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Every such a matrix can be realized as a torus.
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\end{theorem}
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\begin{proof}
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\begin{enumerate}[label={(\Roman*)}]
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\item
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Geometric reason
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\begin{align*}
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\phi_t:
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S^1 \times S^1 &\longrightarrow S^1 \times S^1 \\
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S^1 \times \pt &\longrightarrow \pt \times S^1 \\
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\pt \times S^1 &\longrightarrow S^1 \times \pt \\
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(x, y) & \mapsto (-y, x)
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\end{align*}
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\item
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\end{enumerate}
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\end{proof}
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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/dehn_twist.pdf_tex}}
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\caption{Dehn twist.}
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\label{fig:dehn_twist}
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}
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\end{figure}
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\begin{proof}[Sketch of proof]
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We will show that each diffeomorphism is isotopic to $\begin{pmatrix}
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p & q\\
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r & s
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\end{pmatrix}$.
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\begin{equation*}
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\quot{\Diff_+(S^1 \times S^1)}{\Iso(S^1 \times S^1)} = \mcg(S^1 \times S^1) = \Sl(2, \mathbb{Z})
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\end{equation*}
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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{0.4\textwidth}{!}{\input{images/torus_mu_lambda.pdf_tex}}
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\caption{Choice of meridian and longitude.}
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\label{fig:torus_twist}
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}
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\end{figure}
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\end{proof}
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Let $N = D^2 \times S$ be a tubular neighbourhood of a knot $K$. Consider its boundary $\partial N = S^1 \times S^1$. There exists a simple closed curve $\mu \subset \partial N$ (a meridian) that bounds a disk in $N$. We choose another simple closed curve $\lambda$ (a longitude) so that $\Lk(\lambda, K) = 0$. \\
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????????
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\\
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$\lambda \mu = 1 $ intersection\\
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$\pi_0 (\Gl(2, \mathbb{R})$\\
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???????????
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\\
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In other words a homotopy class: $[\lambda] = 0$ in $H_1(S^3 \setminus N, \mathbb{Z})$.
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