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file 'satellite.pdf' (pdf, eps, ps) +%% +%% To include the image in your LaTeX document, write +%% \input{.pdf_tex} +%% instead of +%% \includegraphics{.pdf} +%% To scale the image, write +%% \def\svgwidth{} +%% \input{.pdf_tex} +%% instead of +%% \includegraphics[width=]{.pdf} +%% +%% Images with a different path to the parent latex file can +%% be accessed with the `import' package (which may need to be +%% installed) using +%% \usepackage{import} +%% in the preamble, and then including the image with +%% \import{}{.pdf_tex} +%% Alternatively, one can specify +%% \graphicspath{{/}} +%% +%% For more information, please see info/svg-inkscape on CTAN: +%% http://tug.ctan.org/tex-archive/info/svg-inkscape +%% +\begingroup% + \makeatletter% + \providecommand\color[2][]{% + \errmessage{(Inkscape) Color is used for the text in Inkscape, but the package 'color.sty' is not loaded}% + \renewcommand\color[2][]{}% + }% + \providecommand\transparent[1]{% + \errmessage{(Inkscape) Transparency is used (non-zero) for the text in Inkscape, but the package 'transparent.sty' is not loaded}% + \renewcommand\transparent[1]{}% + }% + \providecommand\rotatebox[2]{#2}% + \ifx\svgwidth\undefined% + \setlength{\unitlength}{8424.05411848bp}% + \ifx\svgscale\undefined% + \relax% + \else% + \setlength{\unitlength}{\unitlength * \real{\svgscale}}% + \fi% + \else% + \setlength{\unitlength}{\svgwidth}% + \fi% + \global\let\svgwidth\undefined% + \global\let\svgscale\undefined% + \makeatother% + \begin{picture}(1,0.49088964)% + \put(0.36109826,0.62183275){\color[rgb]{0,0,0}\makebox(0,0)[lt]{\begin{minipage}{0.01757932\unitlength}\raggedright \end{minipage}}}% + \put(0.42889331,0.62471825){\color[rgb]{0,0,0}\makebox(0,0)[lt]{\begin{minipage}{0.01757932\unitlength}\raggedright \end{minipage}}}% + \put(0,0){\includegraphics[width=\unitlength,page=1]{satellite.pdf}}% + \put(0.23032975,0.3332003){\color[rgb]{0,0,0}\makebox(0,0)[lt]{\begin{minipage}{0.02367358\unitlength}\raggedright $\lambda$\end{minipage}}}% + \put(0,0){\includegraphics[width=\unitlength,page=2]{satellite.pdf}}% + \put(0.21066574,0.13578336){\color[rgb]{0,0,0}\makebox(0,0)[lt]{\begin{minipage}{0.04131521\unitlength}\raggedright $\mu$\end{minipage}}}% + \put(0,0){\includegraphics[width=\unitlength,page=3]{satellite.pdf}}% + \put(0.4650512,0.44886999){\color[rgb]{0,0,0}\makebox(0,0)[lt]{\begin{minipage}{0.02273321\unitlength}\raggedright $\mu$\end{minipage}}}% + \end{picture}% +\endgroup% diff --git a/images/satellite.svg b/images/satellite.svg new file mode 100644 index 0000000..fb68a44 --- /dev/null +++ b/images/satellite.svg @@ -0,0 +1,1209 @@ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + image/svg+xml + + + + + + + $\Sigma$ $\Sigma$ + + + + + $\lambda$ + $\mu$ + + + + + + + + + + + + + + + $\mu$ + + + diff --git a/images/satellite.svg.2019_06_19_11_40_05.0.svg b/images/satellite.svg.2019_06_19_11_40_05.0.svg new file mode 100644 index 0000000..43a4126 --- /dev/null +++ b/images/satellite.svg.2019_06_19_11_40_05.0.svg @@ -0,0 +1,1066 @@ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + image/svg+xml + + + + + + + $\Sigma$ + + + $\lambda$ $\mu$ + $K$ + + diff --git a/lec_2.tex b/lec_2.tex index af21196..d63119f 100644 --- a/lec_2.tex +++ b/lec_2.tex @@ -206,10 +206,20 @@ $\Delta_{11n34} \equiv 1$. \subsection{Decomposition of $3$-sphere} We know that $3$ - sphere can be obtained by gluing two solid tori: -$S^3 = \partial D^4 = \partial (D^2 \times D^2) = (D^2 \times S^1) \cup (S^1 \times D^2)$. So the complement of solid torus in $S^3$ is another solid torus.\\ -Analytically it can be describes as follow. -Take $(z_1, z_2) \in \mathbb{C}$ such that $\max(\mid z_1 \mid, \mid z_2\mid) = 1 -$. Define following sets: $S_1 = \{ (z_1, z_2) \in S^3: \mid z_1 \mid = 0\} \cong S^1 \times D^2 $ and $S_2 = \{(z_1, z_2) \in S ^3: \mid z_2 \mid = 1 \} \cong D^2 \times S^1$. The intersection $S_1 \cap S_2 = \{(z_1, z_2): \mid z_1 \mid = \mid z_2 \mid = 1 \} \cong S^1 \times S^1$ +\[ +S^3 = \partial D^4 = \partial (D^2 \times D^2) = (D^2 \times S^1) \cup (S^1 \times D^2). +\] +So the complement of solid torus in $S^3$ is another solid torus.\\ +Analytically it can be describes as follow. \\ +Take $(z_1, z_2) \in \mathbb{C}$ such that ${\max(\mid z_1 \vert, \vert z_2\vert) = 1.} +$ +Define following sets: +\begin{align*} +S_1 = \{ (z_1, z_2) \in S^3: \vert z_1 \vert = 0\} \cong S^1 \times D^2 ,\\ +S_2 = \{(z_1, z_2) \in S ^3: \vert z_2 \vert = 1 \} \cong D^2 \times S^1. +\end{align*} +The intersection +$S_1 \cap S_2 = \{(z_1, z_2): \vert z_1 \vert = \vert z_2 \vert = 1 \} \cong S^1 \times S^1$ \begin{figure}[h] \centering{ \def\svgwidth{\linewidth} diff --git a/lec_3.tex b/lec_3.tex index e69de29..520baf2 100644 --- a/lec_3.tex +++ b/lec_3.tex @@ -0,0 +1,170 @@ +\subsection{Algebraic knot} +\noindent +Suppose $F: \mathbb{C}^2 \rightarrow \mathbb{C}$ is a polynomial and $F(0) = 0$. Let take small small sphere $S^3$ around zero. This sphere intersect set of roots of $F$ (zero set of $F$) transversally and by the implicit function theorem the intersection is a manifold. +The dimension of sphere is $3$ and $F^{-1}(0)$ has codimension $2$. +So there is a subspace $L$ - compact one dimensional manifold without boundary. +That means that $L$ is a link in $S^3$. +\begin{figure}[h] +\fontsize{40}{10}\selectfont +\centering{ +\def\svgwidth{\linewidth} +\resizebox{0.2\textwidth}{!}{\input{images/milnor_singular.pdf_tex}} +} +\caption{The intersection of a sphere $S^3$ and zero set of polynomial $F$ is a link $L$.} +\label{fig:milnor_singular} +\end{figure} +%ref: Milnor Singular Points of Complex Hypersurfaces +\begin{theorem} + +$L$ is an unknot if and only if +zero is a smooth point, i.e. +$\bigtriangledown F(0) \neq 0$ (provided $S^3$ has a sufficiently small radius). +\end{theorem} +\noindent +Remark: if $S^3$ is large it can happen that $L$ is unlink, but $F^{-1}(0) \cap B^4$ is "complicated". \\ +%Kyle M. Ormsby +\noindent +In other words: if we take sufficiently small sphere, the link is non-trivial if and only if the point $0$ is singular and the isotopy type of the link doesn't depend on the radius of the sphere. +A link obtained is such a way is called an +algebraic link (in older books on knot theory there is another notion of algebraic link with another meaning). +%ref: Eisenbud, D., Neumann, W. +\begin{example} +Let $p$ and $q$ be coprime numbers such that $p1$. \\ +Zero is an isolated singular point ($\bigtriangledown F(0) = 0$). $F$ is quasi - homogeneous polynomial, so the isotopy class of the link doesn't depend on the choice of a sphere. +Consider $S^3 = \{ (z, w) \in \mathbb{C} : \max( \vert z \vert, \vert w \vert ) = \varepsilon$. +The intersection +$F^{-1}(0) \cap S^3$ is a torus $T(p, q)$. +\\??????????????????? +$F(z, w) = z^p - w^q$\\ +.\\ +$F^{-1}(0) = \{t = t^q, w = t^p\}.$ For unknot $t = \max (\vert t\vert ^p, \vert t \vert^q) = \varepsilon$. +\end{example} +as a corollary we see that $K_T^{n, }$ ???? \\ +is not slice unless $m=0$. \\ +$t = re^{i \Theta}, \Theta \in [0, 2\pi], r = \varepsilon^{\frac{i}{p}}$ + +\begin{figure}[h] +\fontsize{40}{10}\selectfont +\centering{ +\def\svgwidth{\linewidth} +\resizebox{0.2\textwidth}{!}{\input{images/polynomial_and_surface.pdf_tex}} +} +\caption{Sa.} +\label{fig:polynomial_and_surface} +\end{figure} +\begin{theorem} +Suppose $L$ is an algebraic link. $L = F^{-1}(0) \cap S^3$. Let +\begin{align*} +&\varphi : S^3 \setminus L \longrightarrow S^1 \\ +&\varphi(z, w) =\frac{F(z, w)}{\vert F(z, w) \vert}\in S^1, \quad (z, w) \notin F^{-1}(0). +\end{align*} +The map $\varphi$ is a locally trivial fibration. +\end{theorem} +???????\\ +$ rh D \varphi \equiv 1$ +\begin{definition} +A map $\Pi : E \longrightarrow B$ is locally trivial fibration with fiber $F$ if for any $b \in B$, there is a neighbourhood $U \subset B$ such that $\Pi^{-1}(U) \cong U \times $ \\ +????????????\\ $\Gamma$ ?????????????\\ +FIGURES\\ +!!!!!!!!!!!!!!!!!!!!!!!!!!\\ +\end{definition} + +\begin{theorem} +The map $j: \mathscr{C} \longrightarrow \mathbb{Z}^{\infty}$ is a surjection that maps ${K_n}$ to a linear independent set. Moreover $\mathscr{C} \cong \mathbb{Z}$ +\end{theorem} +... +\\ +In general $h$ is defined only up to homotopy, but this means that +\[ +h_* : H_1 (F, \mathbb{Z}) \longrightarrow H_1 (F, \mathbb{Z}) +\] +is well defined \\ +???????????\\ map. +\begin{theorem} +\label{thm:F_as_S} +Suppose $S$ is a Seifert matrix associated with $F$ then $h = S^{-1}S^T$. +\end{theorem} +\begin{proof} +TO WRITE REFERENCE!!!!!!!!!!! +%see Arnold Varchenko vol II +%Picard - Lefschetz formula +%Nemeth (Real Seifert forms +\end{proof} +\noindent +Consequences: +\begin{enumerate} +\item +the Alexander polynomial is the characteristic polynomial of $h$: +\[ +\Delta_L (t) = \det (h - t I d) +\] +In particular $\Delta_L $ is monic (i.e. the top coefficient is $\pm 1$), +???????????????? +\item +S is invertible, +\item +$F$ minimize the genus (i.e. $F$ is minimal genus Seifert surface). +\\??????????????????\\ +\end{enumerate} +% +\begin{definition} +A link $L$ is fibered if there exists a map ${\phi: S^3\setminus L \longrightarrow S^1}$ which is locally trivial fibration. +\end{definition} +\noindent +If $L$ is fibered then Theorem \ref{thm:F_as_S} holds and all its consequences. +\begin{problem} +If $K_1$ and $K_2$ are fibered knots, then also $K_1 \# K_2$ is fibered. +\end{problem} +\noindent +?????????????????????\\ +\begin{problem} +Prove that connected sum is well defined:\\ +$\Delta_{K_1 \# K_2} = +\Delta_{K_1} + \Delta_{K_2}$ and +$g_3(K_1 \# K_2) = g_3(K_1) + g_3(K_2)$. + +\end{problem} +\begin{figure}[h] +\fontsize{12}{10}\selectfont +\centering{ +\def\svgwidth{\linewidth} +\resizebox{1\textwidth}{!}{\input{images/satellite.pdf_tex}} +} +\caption{Whitehead double satellite knot. Its pattern knot embedded non-trivially in an unknotted solid torus $T$ (e.i. $K \not\subset S^3\subset T$) and pattern in a companion knot.} +\label{fig:sattelite} +\end{figure} +\noindent +\subsection{Alternating knot} +\begin{definition} +A knot (link) is called alternating if it admits an alternating diagram. +\end{definition} + +\begin{example} +Figure eight knot is an alternating knot. \hfill\\ +\includegraphics[width=0.5\textwidth]{figure8.png} +\end{example} +\begin{definition} +A reducible crossing in a knot diagram is a crossing for which we can find a circle such that its intersection with a knot diagram is exactly that crossing. A knot diagram without reducible crossing is called reduced. +\end{definition} +\begin{fact} +Any reduced alternating diagram has minimal number of crossings. +\end{fact} +\begin{definition} +The writhe of the diagram is the difference between the number of positive and negative crossings. +\end{definition} +\begin{fact}[Tait] +Any two diagrams of the same alternating knot have the same writhe. +\end{fact} +\begin{fact} +An alternating knot has Alexander polynomial of the form: +$ +a_1t^{n_1} + a_2t^{n_2} + \dots + a_s t^{n_s} +$, where $n_1 < n_2 < \dots < n_s$ and $a_ia_{i+1} < 0$. +\end{fact} +\begin{problem}[open] +What is the minimal $\alpha \in \mathbb{R}$ such that if $z$ is a root of the Alexander polynomial of an alternating knot, then $\Re(z) > \alpha$.\\ +Remark: alternating knots have very simple knot homologies. +\end{problem} +\begin{proposition} +If $T_{p, q}$ is a torus knot, $p < q$, then it is alternating if and only if $p=2$. +\end{proposition} \ No newline at end of file diff --git a/lec_4.tex b/lec_4.tex index 7d0771d..3a0a316 100644 --- a/lec_4.tex +++ b/lec_4.tex @@ -156,6 +156,7 @@ Let $V = \det (tV - V^T) = \det (tA - B^T) - \det(tB - A^T) \end{align*} \begin{corollary} +\label{cor:slice_alex} If $K$ is a slice knot then there exists $f \in \mathbb{Z}[t^{\pm 1}]$ such that $\Delta_K(t) = f(t) \cdot f(t^{-1})$. \end{corollary} \begin{example} diff --git a/lec_5.tex b/lec_5.tex new file mode 100644 index 0000000..43aa11b --- /dev/null +++ b/lec_5.tex @@ -0,0 +1,77 @@ +\begin{theorem} +If $K$ is slice, +then $\sigma_K(t) + = \sign ( (1 - t)S +(1 - \bar{t})S^T)$ +is zero except possibly of finitely many points and $\sigma_K(-1) = \sign(S + S^T) \neq 0$. +\end{theorem} +\begin{proof} +\begin{lemma} +\label{lem:metabolic} +If $V$ is a Hermitian matrix ($\bar{V} = V^T$), $V$ is of size $2n \times 2n$ and +$ +V = \begin{pmatrix} +0 & A \\ +\bar{A}^T & B +\end{pmatrix} +$ and $\det V \neq 0$ then $\sigma(V) = 0$. +\end{lemma} +\begin{definition} +A Hermitian form $V$ is metabolic if $V$ has structure +$\begin{pmatrix} +0 & A\\ +\bar{A}^T & B +\end{pmatrix}$ with half-dimensional null-space. +\end{definition} +\noindent +In other words: non-degenerate metabolic hermitian form has vanishing signature.\\ +We note that $\det(S + S^T) \neq 0$. Hence $\det ( (1 - t) S + (1 - \bar{t})S^T)$ is not identically zero on $S^1$, so it is non-zero except possibly at finitely many points. We apply the Lemma \ref{lem:metabolic}. \\ +Let $t \in S^1 \setminus \{1\}$. Then: +\begin{align*} +&\det((1 - t) S + (1 - \bar{t}) S^T) = +\det((1 - t) S + (t\bar{t} - \bar{t}) S^T) =\\ +&\det((1 - t) (S - \bar{t} - S^T)) = +\det((1 -t)(S - \bar{t} S^T)). +\end{align*} +As $\det (S + S^T) \neq 0$, so $S - \bar{t}S^T \neq 0$. +\end{proof} +?????????????????s\\ +\begin{corollary} +If $K \sim K^\prime$ then for all but finitely many $t \in S^1 \setminus \{1\}: \sigma_K(t) = \sigma_{K^\prime}(t)$. +\end{corollary} +\begin{proof} +If $ K \sim K^\prime$ then $K \# K^\prime$ is slice. +\[ +\sigma_{-K^\prime}(t) = -\sigma_{K^\prime}(t) +\] +\\??????????????\\ +The signature give a homomorphism from the concordance group to $\mathbb{Z}$.\\ +??????????????????\\ +Remark: if $t \in S^1$ is not algebraic over $\mathbb{Z}$, then $\sigma_K(t) \neq 0$ +(we can is the argument that $\mathscr{C} \longrightarrow \mathbb{Z}$ as well). +\end{proof} +\begin{figure}[h] +\fontsize{20}{10}\selectfont +\centering{ +\def\svgwidth{\linewidth} +\resizebox{0.5\textwidth}{!}{\input{images/genus_2_bordism.pdf_tex}} +} +\caption{$K$ and $K^\prime$ are connected by a genus $g$ surface of genus.}\label{fig:genus_2_bordism} +\end{figure} +???????????????????????\\ +\begin{proposition}[Kawauchi inequality] +If there exists a genus $g$ surface as in Figure \ref{fig:genus_2_bordism} +then for almost all $t \in S^1 \setminus \{1\}$ we have $\vert \sigma_K(t) - \sigma_{K^\prime}(t) \vert \leq 2 g$. +\end{proposition} +% Kawauchi Chapter 12 ??? +\begin{lemma} +If $K$ bounds a genus $g$ surface $X \in B^4$ and $S$ is a Seifert form then ${S \in M_{2n \times 2n}}$ has a block structure $\begin{pmatrix} +0 & A\\ +B & C +\end{pmatrix}$, where $0$ is $(n - g) \times (n - g)$ submatrix. +\end{lemma} + +\begin{definition} +The (smooth) four genus $g_4(K)$ is the minimal genus of the surface $\Sigma \in B^4$ such that $\Sigma$ is compact, orientable and $\partial \Sigma = K$. +\end{definition} +\noindent +Remark: $3$ - genus is additive under taking connected sum, but $4$ - genus is not. diff --git a/lec_6.tex b/lec_6.tex new file mode 100644 index 0000000..e69de29 diff --git a/lec_7.tex b/lec_7.tex new file mode 100644 index 0000000..e69de29 diff --git a/lec_8.tex b/lec_8.tex new file mode 100644 index 0000000..e69de29 diff --git a/lec_9.tex b/lec_9.tex new file mode 100644 index 0000000..e69de29 diff --git a/lectures_on_knot_theory.tex b/lectures_on_knot_theory.tex index 10b4152..89d255e 100644 --- a/lectures_on_knot_theory.tex +++ b/lectures_on_knot_theory.tex @@ -123,102 +123,16 @@ \input{lec_2.tex} %add Hurewicz theorem? + \section{\hfill\DTMdate{2019-03-11}} \input{lec_3.tex} -\begin{example} -\begin{align*} -&F: \mathbb{C}^2 \rightarrow \mathbb{C} \text{ a polynomial} \\ -&F(0) = 0 -\end{align*} -\end{example} -\begin{figure}[h] -\fontsize{40}{10}\selectfont -\centering{ -\def\svgwidth{\linewidth} -\resizebox{0.2\textwidth}{!}{\input{images/milnor_singular.pdf_tex}} -} -%\caption{$\mu$ is a meridian and $\lambda$ is a longitude.} -\label{fig:milnor_singular} -\end{figure} -???????????? -\\ -$L$ is a link in $S^3$ \\ -\\?????????????????\\ -$L$ is an unknot if and only if $F(0) \neq 0$ (provided $S^3$ has a sufficiently small radius. -\\ -\noindent -Remark: if $S^3$ is large it can happen that $L$ is unlink, but $F^{-1} \cap B^4$ is "complicated". \\ -????????????\\ -\noindent -\begin{example} -Let $p$ and $q$ be coprime numbers such that $p1$. -\\ -$F^{-1}(0) \cap S^3$ is a solid torus $T(p, q)$. \\ -$F(z, w) = z^p - w^q$\\ -Consider $S^3 = \{ (z, w) \in \mathbb{C} : \max( \mid z \mid, \mid w \mid ) = \varepsilon$.\\ -$F^{-1}(0) = \{t = t^q, w = t^p\}.$ For unknot $t = \max (\mid t\mid ^p, \mid t \mid^q) = \varepsilon$. -\end{example} -as a corollary we see that $K_T^{n, }$ ???? \\ -is not slice unless $m=0$. \\ -$t = re^{i \Theta}, \Theta \in [0, 2\pi], r = \varepsilon^{\frac{i}{p}}$ ?????????????????????????\\ -Suppose $L$ is a diagonal link. $L = F^{-1}(0) \cap S^3$. -\begin{theorem} -The map $j: \mathscr{C} \longrightarrow \mathbb{Z}^{\infty}$ is a surjection that maps ${K_n}$ to a linear independent set. Moreover $\mathscr{C} \cong \mathbb{Z}$ -\end{theorem} - - -\begin{fact}[Milnor Singular Points of Complex Hypersurfaces] -\end{fact} -%\end{comment} -\noindent -An oriented knot is called negative amphichiral if the mirror image $m(K)$ of $K$ is equivalent the reverse knot of $K$: $K^r$. \\ -\begin{problem} -Prove that if $K$ is negative amphichiral, then $K \# K = 0$ in -$\mathscr{C}$. -% -%\\ -%Hint: $ -K = m(K)^r = (K^r)^r = K$ -\end{problem} -\begin{example} -Figure 8 knot is negative amphichiral. -\end{example} -% -% -% -\begin{definition} -A link $L$ is fibered if there exists a map ${\phi: S^3\setminus L \longleftarrow S^1}$ which is locally trivial fibration. -\end{definition} - - \section{Concordance group \hfill\DTMdate{2019-03-18}} \input{lec_4.tex} - \section{\hfill\DTMdate{2019-03-25}} -\begin{theorem} -If $K$ is slice, -then $\sigma_K(t) - = \sign ( (1 - t)S +(1 - \bar{t})S^T)$ -is zero except possibly of finitely many points and $\sigma_K(-1) = \sign(S + S^T) \neq 0$. -\end{theorem} -\begin{proof} -\begin{lemma} -If $V$ is a Hermitian matrix ($\bar{V} = V^T$), $V$ is of size $2n \times 2n$ and -$ -V = \begin{pmatrix} -0 & A \\ -\bar{A}^T & B -\end{pmatrix} -$ -\end{lemma} -\end{proof} -\begin{definition} -The (smooth) four genus $g_4(K)$ is the minimal genus of the surface $\Sigma \in B^4$ such that $\Sigma$ is compact, orientable and $\partial \Sigma = K$. -\end{definition} -\noindent -Remark: $3$ - genus is additive under taking connected sum, but $4$ - genus is not. +\input{lec_5.tex} \section{\hfill\DTMdate{2019-04-08}} % @@ -750,6 +664,24 @@ For knots the order of the Alexander module is the Alexander polynomial. $M$ is well defined up to a unit in $R$. \subsection*{Blanchfield pairing} \section{balagan} + +\begin{fact}[Milnor Singular Points of Complex Hypersurfaces] +\end{fact} +%\end{comment} +\noindent +An oriented knot is called negative amphichiral if the mirror image $m(K)$ of $K$ is equivalent the reverse knot of $K$: $K^r$. \\ +\begin{problem} +Prove that if $K$ is negative amphichiral, then $K \# K = 0$ in +$\mathscr{C}$. +% +%\\ +%Hint: $ -K = m(K)^r = (K^r)^r = K$ +\end{problem} +\begin{example} +Figure 8 knot is negative amphichiral. +\end{example} +% +% \begin{theorem} Let $H_p$ be a $p$ - torsion part of $H$. There exists an orthogonal decomposition of $H_p$: \[