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+
+
+
+
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@@ -0,0 +1,63 @@
+%% Creator: Inkscape inkscape 0.92.2, www.inkscape.org
+%% PDF/EPS/PS + LaTeX output extension by Johan Engelen, 2010
+%% Accompanies image file 'genus_2_bordism.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
+%%
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+
+
+
+
diff --git a/images/polynomial_and_surface.pdf b/images/polynomial_and_surface.pdf
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diff --git a/images/polynomial_and_surface.pdf_tex b/images/polynomial_and_surface.pdf_tex
new file mode 100644
index 0000000..788eee7
--- /dev/null
+++ b/images/polynomial_and_surface.pdf_tex
@@ -0,0 +1,67 @@
+%% Creator: Inkscape inkscape 0.92.2, www.inkscape.org
+%% PDF/EPS/PS + LaTeX output extension by Johan Engelen, 2010
+%% Accompanies image file 'polynomial_and_surface.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
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+ \put(1.67978829,0.06708039){\color[rgb]{0,0,0}\makebox(0,0)[lt]{\begin{minipage}{0.54940718\unitlength}\raggedright \end{minipage}}}%
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diff --git a/images/polynomial_and_surface.svg b/images/polynomial_and_surface.svg
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--- /dev/null
+++ b/images/polynomial_and_surface.svg
@@ -0,0 +1,239 @@
+
+
+
+
diff --git a/images/satellite.pdf b/images/satellite.pdf
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diff --git a/images/satellite.pdf_tex b/images/satellite.pdf_tex
new file mode 100644
index 0000000..4e1e065
--- /dev/null
+++ b/images/satellite.pdf_tex
@@ -0,0 +1,61 @@
+%% Creator: Inkscape inkscape 0.92.2, www.inkscape.org
+%% PDF/EPS/PS + LaTeX output extension by Johan Engelen, 2010
+%% Accompanies image 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
+%%
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+ \makeatletter%
+ \providecommand\color[2][]{%
+ \errmessage{(Inkscape) Color is used for the text in Inkscape, but the package 'color.sty' is not loaded}%
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+ \put(0.23032975,0.3332003){\color[rgb]{0,0,0}\makebox(0,0)[lt]{\begin{minipage}{0.02367358\unitlength}\raggedright $\lambda$\end{minipage}}}%
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+ \put(0.21066574,0.13578336){\color[rgb]{0,0,0}\makebox(0,0)[lt]{\begin{minipage}{0.04131521\unitlength}\raggedright $\mu$\end{minipage}}}%
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+ \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
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--- /dev/null
+++ b/images/satellite.svg
@@ -0,0 +1,1209 @@
+
+
+
+
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
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+++ b/images/satellite.svg.2019_06_19_11_40_05.0.svg
@@ -0,0 +1,1066 @@
+
+
+
+
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 $H_p$:
\[