diff --git a/images/ball_4_pushed_seifert.pdf b/images/ball_4_pushed_seifert.pdf
index dd1e3e8..c46ffa1 100644
Binary files a/images/ball_4_pushed_seifert.pdf and b/images/ball_4_pushed_seifert.pdf differ
diff --git a/images/ball_4_pushed_seifert.pdf_tex b/images/ball_4_pushed_seifert.pdf_tex
index a089784..292170f 100644
--- a/images/ball_4_pushed_seifert.pdf_tex
+++ b/images/ball_4_pushed_seifert.pdf_tex
@@ -56,8 +56,8 @@
\put(0,0){\includegraphics[width=\unitlength,page=2]{ball_4_pushed_seifert.pdf}}%
\put(0.59916734,0.31598118){\color[rgb]{0,0,0}\makebox(0,0)[lt]{\begin{minipage}{0.90744424\unitlength}\raggedright $g(F) = g_4(K)$ \end{minipage}}}%
\put(0.6042564,0.37550965){\color[rgb]{0,0,0}\makebox(0,0)[lt]{\begin{minipage}{0.85985262\unitlength}\raggedright $F \subset B^4$ \end{minipage}}}%
- \put(0.6549243,0.82850612){\color[rgb]{0,0,0}\makebox(0,0)[lt]{\begin{minipage}{1.03767986\unitlength}\raggedright $S^3$ \end{minipage}}}%
+ \put(0.72361375,0.82850612){\color[rgb]{0,0,0}\makebox(0,0)[lt]{\begin{minipage}{1.03767986\unitlength}\raggedright $S^3$ \end{minipage}}}%
\put(0,0){\includegraphics[width=\unitlength,page=3]{ball_4_pushed_seifert.pdf}}%
- \put(0.81311761,0.76290458){\color[rgb]{0,0,0}\makebox(0,0)[lt]{\begin{minipage}{0.30641817\unitlength}\raggedright $\Sigma$\end{minipage}}}%
+ \put(0.87332011,0.71316438){\color[rgb]{0,0,0}\makebox(0,0)[lt]{\begin{minipage}{0.20662167\unitlength}\raggedright $\Sigma$\end{minipage}}}%
\end{picture}%
\endgroup%
diff --git a/images/ball_4_pushed_seifert.svg b/images/ball_4_pushed_seifert.svg
index b40c812..183dbfe 100644
--- a/images/ball_4_pushed_seifert.svg
+++ b/images/ball_4_pushed_seifert.svg
@@ -1538,13 +1538,13 @@
borderopacity="1.0"
inkscape:pageopacity="0.0"
inkscape:pageshadow="2"
- inkscape:zoom="1.3101783"
- inkscape:cx="251.59804"
- inkscape:cy="115.99934"
+ inkscape:zoom="2.2812777"
+ inkscape:cx="159.21118"
+ inkscape:cy="140.71061"
inkscape:document-units="px"
inkscape:current-layer="layer1"
showgrid="false"
- inkscape:window-width="1393"
+ inkscape:window-width="1392"
inkscape:window-height="855"
inkscape:window-x="0"
inkscape:window-y="1"
@@ -1624,33 +1624,30 @@
style="font-size:40px;line-height:1.25">$\Sigma$
-
-
-
-
-
-
+
+
+
+
$F \subset B^4$
+ crossing_points_vector="113.40108 | 63.222085 | 0 | 0 | 0 | 5 | 0.6312852 | 6.3026953 | 1 | 82.948274 | 61.432802 | 0 | 0 | 1 | 4 | 1.060697 | 4.7438172 | 1 | 98.060004 | 143.61611 | 0 | 0 | 2 | 7 | 2.6579504 | 7.5171585 | -1 | 98.343835 | 80.092075 | 0 | 0 | 3 | 6 | 4.2295883 | 6.6117698 | -1" />
+ inkscape:window-width="1397"
+ inkscape:window-height="855"
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+ inkscape:window-y="1"
+ inkscape:window-maximized="1"
+ fit-margin-top="0"
+ fit-margin-left="0"
+ fit-margin-right="0"
+ fit-margin-bottom="0" />
@@ -62,14 +66,17 @@
+ id="layer1"
+ transform="translate(-37.022916,-50.153379)">
+ sodipodi:nodetypes="ssssssssss"
+ inkscape:export-xdpi="90"
+ inkscape:export-ydpi="90" />
diff --git a/images/satellite.pdf b/images/satellite.pdf
index 0fa5a0c..0b258a4 100644
Binary files a/images/satellite.pdf and b/images/satellite.pdf differ
diff --git a/images/satellite.pdf_tex b/images/satellite.pdf_tex
index 4e1e065..25a4a55 100644
--- a/images/satellite.pdf_tex
+++ b/images/satellite.pdf_tex
@@ -52,10 +52,10 @@
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\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(0.23032974,0.3332003){\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}}}%
+ \put(0.46450745,0.42798995){\color[rgb]{0,0,0}\makebox(0,0)[lt]{\begin{minipage}{0.0227986\unitlength}\raggedright $\mu$\end{minipage}}}%
\end{picture}%
\endgroup%
diff --git a/images/satellite.svg b/images/satellite.svg
index fb68a44..9e11cce 100644
--- a/images/satellite.svg
+++ b/images/satellite.svg
@@ -29,7 +29,7 @@
add_stroke_width="true"
add_other_stroke_width="true"
switcher_size="15"
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+ crossing_points_vector="1521.3825 | 1359.9656 | 3 | 4 | 0 | 0 | 6 | 0 | -1 | 1495.213 | 1248.9664 | 4 | 5 | 1 | 0 | 1 | 2.7869328e-13 | 1" />
+ crossing_points_vector="1313.5623 | 805.02533 | 0 | 0 | 0 | 3 | 8.4104594 | 18.279706 | -1 | 829.65847 | 898.34753 | 0 | 0 | 1 | 4 | 9.3663943 | 23.295043 | -1 | 326.55758 | 1007.7342 | 0 | 0 | 2 | 5 | 10.326548 | 27.465011 | -1" />
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diff --git a/images/torus_alpha_beta.pdf b/images/torus_alpha_beta.pdf
new file mode 100644
index 0000000..a744757
Binary files /dev/null and b/images/torus_alpha_beta.pdf differ
diff --git a/images/torus_alpha_beta.pdf_tex b/images/torus_alpha_beta.pdf_tex
new file mode 100644
index 0000000..3a51b47
--- /dev/null
+++ b/images/torus_alpha_beta.pdf_tex
@@ -0,0 +1,59 @@
+%% Creator: Inkscape inkscape 0.92.2, www.inkscape.org
+%% PDF/EPS/PS + LaTeX output extension by Johan Engelen, 2010
+%% Accompanies image file 'torus_alpha_beta.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(0.63089219,0.39833514){\color[rgb]{0,0,0}\makebox(0,0)[lt]{\begin{minipage}{0.38670889\unitlength}\raggedright ${\alpha \cdot \beta = - \beta \cdot \alpha}$\\ \end{minipage}}}%
+ \put(0,0){\includegraphics[width=\unitlength,page=1]{torus_alpha_beta.pdf}}%
+ \put(0.94608607,0.30060216){\color[rgb]{0,0,0}\makebox(0,0)[lt]{\begin{minipage}{0.38670889\unitlength}\raggedright $\beta$\\ \end{minipage}}}%
+ \put(0.44999409,0.1665571){\color[rgb]{0,0,0}\makebox(0,0)[lt]{\begin{minipage}{0.38670889\unitlength}\raggedright $\alpha$\\ \end{minipage}}}%
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+ \end{picture}%
+\endgroup%
diff --git a/images/torus_alpha_beta.svg b/images/torus_alpha_beta.svg
index 2bed741..1268d87 100644
--- a/images/torus_alpha_beta.svg
+++ b/images/torus_alpha_beta.svg
@@ -11,15 +11,176 @@
xmlns:xlink="http://www.w3.org/1999/xlink"
xmlns:sodipodi="http://sodipodi.sourceforge.net/DTD/sodipodi-0.dtd"
xmlns:inkscape="http://www.inkscape.org/namespaces/inkscape"
- width="210mm"
- height="297mm"
- viewBox="0 0 210 297"
+ width="150.85768mm"
+ height="62.708881mm"
+ viewBox="0 0 150.85768 62.708881"
version="1.1"
id="svg4590"
inkscape:version="0.92.2 5c3e80d, 2017-08-06"
sodipodi:docname="torus_alpha_beta.svg">
+
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+
+
+
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+
+
+
+
+
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@@ -970,7 +1145,8 @@
+ id="layer1"
+ transform="translate(-18.706472,-63.135373)">
$\Sigma$ genus $3$ ${\alpha \cdot \beta = - \beta \cdot \alpha}$
+
+
+ $\beta$ $\alpha$
+
+
+
+
+
+
+
+
diff --git a/lec_2.tex b/lec_2.tex
index d63119f..94dbb4a 100644
--- a/lec_2.tex
+++ b/lec_2.tex
@@ -211,7 +211,7 @@ S^3 = \partial D^4 = \partial (D^2 \times D^2) = (D^2 \times S^1) \cup (S^1 \ti
\]
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.}
+Take $(z_1, z_2) \in \mathbb{C}$ such that ${\max(\vert z_1 \vert, \vert z_2\vert) = 1.}
$
Define following sets:
\begin{align*}
@@ -219,7 +219,7 @@ 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
-$S_1 \cap S_2 = \{(z_1, z_2): \vert z_1 \vert = \vert z_2 \vert = 1 \} \cong S^1 \times S^1$
+$S_1 \cap S_2 = \{(z_1, z_2): \vert z_1 \vert = \vert z_2 \vert = 1 \} \cong S^1 \times S^1$.
\begin{figure}[h]
\centering{
\def\svgwidth{\linewidth}
@@ -256,7 +256,9 @@ Suppose $K \subset S^3$ and $\pi_1(S^3 \setminus K)$ is infinite cyclic ($\mathb
\end{corollary}
\begin{proof}
Let $N$ be a tubular neighbourhood of a knot $K$ and $M = S^3 \setminus N$ its complement. Then $\partial M = S^1 \times S^1$. Let $f : \pi_1(\partial M ) \longrightarrow \pi_1(M)$.
-If $\pi_1(M)$ is infinite cyclic group then the map $f$ is non-trivial. Suppose ${\lambda \in \ker (\pi_1(S^1 \times S^1) \longrightarrow \pi_1(M)}$. There is a map $g: (D^2, \partial D^2) \longrightarrow (M, \partial M)$ such that $g(\partial D^2) = \lambda$. By Dehn's lemma there exists an embedding ${h: (D^2, \partial D^2) \longhookrightarrow (M, \partial M)}$ such that
+If $\pi_1(M)$ is infinite cyclic group then the map $f$ is non-trivial. Suppose ${\lambda \in \ker (\pi_1(S^1 \times S^1) \longrightarrow \pi_1(M)}$.
+There is a map $g: (D^2, \partial D^2) \longrightarrow (M, \partial M)$ such that $g(\partial D^2) = \lambda$.\\
+ By Dehn's lemma there exists an embedding ${h: (D^2, \partial D^2) \longhookrightarrow (M, \partial M)}$ such that
$h\big|_{\partial D^2} = f \big|_{\partial D^2}$ and $h(\partial D^2) = \lambda$.
Let $\Sigma$ be a union of the annulus and the image of $\partial D^2$.
\\???? $g_3$?\\
diff --git a/lec_3.tex b/lec_3.tex
index 520baf2..2aef0ee 100644
--- a/lec_3.tex
+++ b/lec_3.tex
@@ -92,7 +92,7 @@ TO WRITE REFERENCE!!!!!!!!!!!
\end{proof}
\noindent
Consequences:
-\begin{enumerate}
+\begin{enumerate}[label={(\arabic*)}]
\item
the Alexander polynomial is the characteristic polynomial of $h$:
\[
@@ -139,10 +139,16 @@ $g_3(K_1 \# K_2) = g_3(K_1) + g_3(K_2)$.
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{figure}[h]
+\fontsize{12}{10}\selectfont
+\centering{
+\def\svgwidth{\linewidth}
+\includegraphics[width=0.3\textwidth]{figure8.png}
+}
+\caption{Example: figure eight knot is an alternating knot.}
+\label{fig:fig8}
+\end{figure}
+
\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}
diff --git a/lec_5.tex b/lec_5.tex
index 43aa11b..56128a2 100644
--- a/lec_5.tex
+++ b/lec_5.tex
@@ -5,6 +5,8 @@ then $\sigma_K(t)
is zero except possibly of finitely many points and $\sigma_K(-1) = \sign(S + S^T) \neq 0$.
\end{theorem}
\begin{proof}
+\noindent
+We will use the following lemma.
\begin{lemma}
\label{lem:metabolic}
If $V$ is a Hermitian matrix ($\bar{V} = V^T$), $V$ is of size $2n \times 2n$ and
@@ -34,7 +36,6 @@ Let $t \in S^1 \setminus \{1\}$. Then:
\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}
@@ -69,9 +70,76 @@ If $K$ bounds a genus $g$ surface $X \in B^4$ and $S$ is a Seifert form then ${S
B & C
\end{pmatrix}$, where $0$ is $(n - g) \times (n - g)$ submatrix.
\end{lemma}
-
+??????????????????????\\
+\begin{align*}
+\dim H_1(Z) = 2 n\\
+\dim H_1 (Y) = 2 n + 2 g\\
+\dim (\ker (H_1, Y) \longrightarrow H_1(\Omega)) = n + g\\
+Y = X \sum \Sigma
+\end{align*}
+\noindent
+If $\alpha, \beta \in \ker(H_1(\Sigma \longrightarrow H_1(\Omega))$, then ${\Lk(\alpha, \beta^+) = 0}$.
+\begin{corollary}
+If $t$ is nota ???? of $\det $ ????
+then $\vert \sigma_K(t) \vert \leq 2g$.\\
+\end{corollary}
+\noindent
+If there exists cobordism of genus $g$ between $K$ and $K^\prime$ like shown in Figure \ref{fig:genus_2_bordism}, then $K \# -K^\prime$ bounds a surface of genus $g$ in $B^4$.
\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.
+\begin{example}
+\begin{itemize}
+\item Let $K = T(2, 3)$. $\sigma(K) = -2$, therefore $T(2, 3)$ isn't a slice knot.
+\item Let $K$ be a trefoil and $K^\prime$ a mirror of a trefoil. $g_4(K^\prime) = 1$, but $g_(K \# K^\prime) = 0$.
+\\?????????????????????\\
+\item
+?????????????\\
+The equality:
+\[
+g_4(T(p, q) ) = \frac{1}{2} (p - 1) (g -1)
+\]
+was conjecture in the '70 and proved by P. Kronheimer and T. Mrówka.
+\end{itemize}
+\end{example}
+\begin{proposition}
+$g_4 (T(p, q) \# -T(r, s))$ is in general hopelessly unknown.
+\\???????????????\\
+essentially $\sup \vert \sigma_K(t) \vert \leq 2 g_n(K)$
+\end{proposition}
+\begin{definition}
+A knot $K$ is called topologically slice if $K$ bounds a topological locally flat disc in $B^4$ (it has tubular neighbourhood).
+\end{definition}
+\begin{theorem}[Freedman, '82]
+If $\Delta_K(t) \geq 1$, then $K$ is topologically slice, but not necessarily smoothly slice.
+\end{theorem}
+\begin{theorem}[Powell, 2015]
+If $K$ is genus g
+\\(top. loc.?????????)\\
+cobordant to $K^\prime$,
+then $\vert \sigma_K(t) - \sigma_{K^\prime}(t) \vert \leq 2 g$. \\
+If $g_4^{\mytop}(K) \geq $ ?????ess $\sup \vert \sigma_K(t) \vert$ and ?????????\\
+$\dim \ker (H_1 (Y) \longrightarrow H_1(\Omega)) = \frac{1}{2} \dim H_1(Y)$.
+\end{theorem}
+???????????????
+\[
+H^1 (B^4 \setminus Y, \mathbb{Z}) = [B^4 \setminus Y, S^1]
+\]
+\noindent
+Remark: unless $p=2$ or $p = 3 \wedge q = 4$:
+\[
+g_4^\top (T(p, q)) < q_4(T(p, q))
+\]
+%??????????????????????
+\begin{definition}
+The Witt group $W$ of $\mathbb{Z}[t, t^{-1}]$ elements are classes of non-degenerate
+forms over $\mathbb{Z}[t, t^{-1}]$ under the equivalence relation $V \sim W$ if $V \oplus - W$ is metabolic.
+\end{definition}
+\noindent
+If $S$ differs from $S^\prime$ by a row extension, then
+$(1 - t) S + (1 - \bar{t}^{-1}) S^T$ is Witt equivalence to $(1 - t) S^\prime + (1 - t^{-1})S^T$.
+%???????????????????????????
+\noindent
+A form is meant as hermitian with respect to this involution: $A^T = A: (a, b) = \bar{(a, b)}$.
\ No newline at end of file
diff --git a/lectures_on_knot_theory.tex b/lectures_on_knot_theory.tex
index 89d255e..97e49d5 100644
--- a/lectures_on_knot_theory.tex
+++ b/lectures_on_knot_theory.tex
@@ -85,6 +85,8 @@
\DeclareMathOperator{\Hom}{Hom}
\DeclareMathOperator{\rank}{rank}
\DeclareMathOperator{\ord}{ord}
+\DeclareMathOperator{\mytop}{top}
+
\DeclareMathOperator{\Gl}{GL}
\DeclareMathOperator{\Sl}{SL}
\DeclareMathOperator{\Lk}{lk}
@@ -148,28 +150,84 @@ H_2(X, \mathbb{Z}) \longrightarrow \mathbb{Z}$ - symmetric, non singular.
\\
Let $A$ and $B$ be closed, oriented surfaces in $X$.
\begin{proposition}
-$A \cdot B$ doesn't depend of choice of $A$ and $B$ in their homology classes $[A], [B] \in H_2(X, \mathbb{Z})$. \\
+$A \cdot B$ doesn't depend of choice of $A$ and $B$ in their homology classes:
+\[
+[A], [B] \in H_2(X, \mathbb{Z}).
+\]
+\end{proposition}
\noindent
+\\
+
If $M$ is an $m$ - dimensional close, connected and orientable manifold, then $H_m(M, \mathbb{Z})$ and the orientation if $M$ determined a cycle $[M] \in H_m(M, \mathbb{Z})$, called the fundamental cycle.
\begin{example}
If $\omega$ is an $m$ - form then:
\[
- = [\omega]
+\int_M \omega = [\omega]([M]), \quad [\omega] \in H^m_\Omega(M), \ [M] \in H_m(M).
\]
-\end{example}
+
+\end{example}
+????????????????????????????????????????????????
+\begin{figure}[h]
+\fontsize{20}{10}\selectfont
+\centering{
+\def\svgwidth{\linewidth}
+\resizebox{0.8\textwidth}{!}{\input{images/torus_alpha_beta.pdf_tex}}
+}
+\caption{$\beta$ cross $3$ times the disk bounded by $\alpha$.
+$T_X \alpha + T_X \beta = T_Z \Sigma$
+}\label{fig:torus_alpha_beta}
+\end{figure}
+
+\begin{theorem}
+Any non-degenerate form
+\[
+A : \mathbb{Z}^n \times \mathbb{Z}^n \longrightarrow \mathbb{Z}
+\]
+can be realized as an intersection form of a simple connected $4$-dimensional manifold.
+\end{theorem}
+??????????????????????????
+\begin{theorem}[Donaldson, 1982]
+If $A$ is an even defined intersection form of a smooth $4$-manifold then it is diagonalizable over $\mathbb{Z}$.
+\end{theorem}
+??????????????????????????
+??????????????????????????
+??????????????????????????
+??????????????????????????
+\begin{definition}
+even define
+\end{definition}
+Suppose $X$ us $4$ -manifold with a boundary such that $H_1(X) = 0$.
+
%$A \cdot B$ gives the pairing as ??
- \end{proposition}
\section{\hfill\DTMdate{2019-04-15}}
-In other words:\\
+\begin{theorem}
+Suppose that $K \subset S^3$ is a slice knot (i.e. $K$ bound a disk in $B^4$).
+Then if $F$ is a Seifert surface of $K$ and $V$ denotes the associated Seifet matrix, then there exists $P \in \Gl_g(\mathbb{Z})$ such that:
+\\??????????????? T ????????
+\begin{align}
+PVP^{-1} =
+\begin{pmatrix}
+0 & A\\
+B & C
+\end{pmatrix}, \quad A, C, C \in M_{g \times g} (\mathbb{Z})
+\end{align}
+\end{theorem}
+In other words you can find rank $g$ direct summand $\mathcal{Z}$ of $H_1(F)$ \\
+????????????\\
+such that for any
+$\alpha, \beta \in \mathcal{L}$ the linking number $\Lk (\alpha, \beta^+) = 0$.
+\begin{definition}
+An abstract Seifert matrix (i. e.
+\end{definition}
Choose a basis $(b_1, ..., b_i)$ \\
???\\
of $H_2(Y, \mathbb{Z}$, then $A = (b_i, b_y)$ \\??\\ is a matrix of intersection form:
\begin{align*}
\quot{\mathbb{Z}^n}{A\mathbb{Z}^n} \cong H_1(Y, \mathbb{Z}).
\end{align*}
-In particular $\mid \det A\mid = \# H_1(Y, \mathbb{Z})$.\\
+In particular $\vert \det A\vert = \# H_1(Y, \mathbb{Z})$.\\
That means - what is happening on boundary is a measure of degeneracy.
\begin{center}
@@ -426,7 +484,7 @@ Therefore:
&P^{\prime} = g^{\prime}\overbar{g}
\end{align*}
We set $g = g^{\prime}(t - \zeta)(t - \overbar{\zeta})$ and
-$P = g \overbar{g}$. Suppose $\zeta \in S^1$. Then $(t - \zeta)^2 \mid P$ (at least - otherwise it would change sign). Therefore:
+$P = g \overbar{g}$. Suppose $\zeta \in S^1$. Then $(t - \zeta)^2 \vert P$ (at least - otherwise it would change sign). Therefore:
\begin{align*}
&P^{\prime} = \frac{P}{(t - \zeta)^2(t^{-1} - \zeta)^2}\\
&g = (t - \zeta)(t^{-1} - \zeta) g^{\prime} \quad \text{etc.}