49 lines
1.2 KiB
TeX
49 lines
1.2 KiB
TeX
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\begin{lemma}[Milnor Singular Points of Complex Hypersurfaces]
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\end{lemma}
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%\end{comment}
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\noindent
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An oriented knot is called negative amphichiral if the mirror image $m(K)$ of $K$ is equivalent the reverse knot of $K$: $K^r$. \\
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\begin{problem}
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Prove that if $K$ is negative amphichiral, then $K \# K = 0$ in
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$\mathscr{C}$.
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%
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%\\
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%Hint: $ -K = m(K)^r = (K^r)^r = K$
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\end{problem}
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\begin{example}
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Figure 8 knot is negative amphichiral.
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\end{example}
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%
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%
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\begin{theorem}
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Let $H_p$ be a $p$ - torsion part of $H$. There exists an orthogonal decomposition of $H_p$:
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\[
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H_p = H_{p, 1} \oplus \dots \oplus H_{p, r_p}.
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\]
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$H_{p, i}$ is a cyclic module:
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\[
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H_{p, i} = \quot{\mathbb{Z}[t, t^{-1}]}{p^{k_i} \mathbb{Z} [t, t^{-1}]}
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\]
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\end{theorem}
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\noindent
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The proof is the same as over $\mathbb{Z}$.
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\noindent
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%Add NotePrintSaveCiteYour opinionEmailShare
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%Saveliev, Nikolai
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%Lectures on the Topology of 3-Manifolds
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%An Introduction to the Casson Invariant
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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{0.5\textwidth}{!}{\input{images/ball_4_alpha_beta.pdf_tex}}
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}
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%\caption{Sketch for Fact %%\label{fig:concordance_m}
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\end{figure}
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