diff --git a/images/genus_2_bordism.pdf b/images/genus_2_bordism.pdf
index 6089acb..70ef42e 100644
Binary files a/images/genus_2_bordism.pdf and b/images/genus_2_bordism.pdf differ
diff --git a/images/genus_2_bordism.svg b/images/genus_2_bordism.svg
index 81cb86a..f7e768f 100644
--- a/images/genus_2_bordism.svg
+++ b/images/genus_2_bordism.svg
@@ -946,28 +946,6 @@
offset="0"
style="stop-color:#d40000;stop-opacity:1;" />
-
-
-
-
-
- .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]{}%
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+ \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}(1,0.72919638)%
+ \put(0,0){\includegraphics[width=\unitlength,page=1]{genus_bordism_proof.pdf}}%
+ \put(0.5553397,1.5895786){\color[rgb]{0,0,0}\makebox(0,0)[lt]{\begin{minipage}{0.09784772\unitlength}\raggedright \end{minipage}}}%
+ \put(0,0){\includegraphics[width=\unitlength,page=2]{genus_bordism_proof.pdf}}%
+ \put(0.84677759,0.62354405){\color[rgb]{0,0,0}\makebox(0,0)[lt]{\begin{minipage}{0.18126183\unitlength}\raggedright \end{minipage}}}%
+ \put(0.88638971,0.57742432){\color[rgb]{0,0,0}\makebox(0,0)[lt]{\begin{minipage}{0.07636904\unitlength}\raggedright \end{minipage}}}%
+ \put(0.03411145,0.64157429){\color[rgb]{0,0,0}\makebox(0,0)[lt]{\begin{minipage}{0.10538925\unitlength}\raggedright \end{minipage}}}%
+ \put(0,0){\includegraphics[width=\unitlength,page=3]{genus_bordism_proof.pdf}}%
+ \put(0.02081414,0.65430801){\color[rgb]{0,0,0}\makebox(0,0)[lt]{\begin{minipage}{0.0453964\unitlength}\raggedright $K$\end{minipage}}}%
+ \put(0.89011285,0.69804096){\color[rgb]{0,0,0}\makebox(0,0)[lt]{\begin{minipage}{0.13452011\unitlength}\raggedright $K\prime$\end{minipage}}}%
+ \put(0,0){\includegraphics[width=\unitlength,page=4]{genus_bordism_proof.pdf}}%
+ \put(-0.0012116,0.28170585){\color[rgb]{0,0,0}\makebox(0,0)[lt]{\begin{minipage}{0.0453964\unitlength}\raggedright $K$\end{minipage}}}%
+ \put(0.8680871,0.32543879){\color[rgb]{0,0,0}\makebox(0,0)[lt]{\begin{minipage}{0.13452011\unitlength}\raggedright $K\prime$\end{minipage}}}%
+ \put(0,0){\includegraphics[width=\unitlength,page=5]{genus_bordism_proof.pdf}}%
+ \end{picture}%
+\endgroup%
diff --git a/images/genus_bordism_proof.svg b/images/genus_bordism_proof.svg
new file mode 100644
index 0000000..7b7f4fa
--- /dev/null
+++ b/images/genus_bordism_proof.svg
@@ -0,0 +1,1276 @@
+
+
+
+
diff --git a/images/genus_bordism_zeros.pdf b/images/genus_bordism_zeros.pdf
new file mode 100644
index 0000000..e83d8b1
Binary files /dev/null and b/images/genus_bordism_zeros.pdf differ
diff --git a/images/genus_bordism_zeros.pdf_tex b/images/genus_bordism_zeros.pdf_tex
new file mode 100644
index 0000000..8144dbe
--- /dev/null
+++ b/images/genus_bordism_zeros.pdf_tex
@@ -0,0 +1,60 @@
+%% Creator: Inkscape inkscape 0.92.2, www.inkscape.org
+%% PDF/EPS/PS + LaTeX output extension by Johan Engelen, 2010
+%% Accompanies image file 'genus_bordism_zeros.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]{%
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+ \renewcommand\transparent[1]{}%
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+ \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}(1,0.87333181)%
+ \put(0,0){\includegraphics[width=\unitlength,page=1]{genus_bordism_zeros.pdf}}%
+ \put(1.27431841,1.84948009){\color[rgb]{0,0,0}\makebox(0,0)[lt]{\begin{minipage}{0.39774488\unitlength}\raggedright \end{minipage}}}%
+ \put(1.36004492,1.65392014){\color[rgb]{0,0,0}\makebox(0,0)[lt]{\begin{minipage}{0.39774488\unitlength}\raggedright \end{minipage}}}%
+ \put(0,0){\includegraphics[width=\unitlength,page=2]{genus_bordism_zeros.pdf}}%
+ \put(0.6042564,0.37550965){\color[rgb]{0,0,0}\makebox(0,0)[lt]{\begin{minipage}{0.85985262\unitlength}\raggedright $X \subset B^4$ \end{minipage}}}%
+ \put(0,0){\includegraphics[width=\unitlength,page=3]{genus_bordism_zeros.pdf}}%
+ \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/genus_bordism_zeros.svg b/images/genus_bordism_zeros.svg
new file mode 100644
index 0000000..2c44009
--- /dev/null
+++ b/images/genus_bordism_zeros.svg
@@ -0,0 +1,1684 @@
+
+
+
+
diff --git a/images/intersection_form_A_B.pdf b/images/intersection_form_A_B.pdf
new file mode 100644
index 0000000..fc1b6d3
Binary files /dev/null and b/images/intersection_form_A_B.pdf differ
diff --git a/images/intersection_form_A_B.pdf_tex b/images/intersection_form_A_B.pdf_tex
new file mode 100644
index 0000000..6f98985
--- /dev/null
+++ b/images/intersection_form_A_B.pdf_tex
@@ -0,0 +1,55 @@
+%% Creator: Inkscape inkscape 0.92.2, www.inkscape.org
+%% PDF/EPS/PS + LaTeX output extension by Johan Engelen, 2010
+%% Accompanies image file 'intersection_form_A_B.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%
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+ \setlength{\unitlength}{\svgwidth}%
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+ \global\let\svgscale\undefined%
+ \makeatother%
+ \begin{picture}(1,1.41428571)%
+ \put(0,0){\includegraphics[width=\unitlength,page=1]{intersection_form_A_B.pdf}}%
+ \put(0.47876984,1.29411848){\color[rgb]{0,0,0}\makebox(0,0)[lt]{\begin{minipage}{0.13859127\unitlength}\raggedright $A$\end{minipage}}}%
+ \end{picture}%
+\endgroup%
diff --git a/images/intersection_form_A_B.svg b/images/intersection_form_A_B.svg
new file mode 100644
index 0000000..f7f06a1
--- /dev/null
+++ b/images/intersection_form_A_B.svg
@@ -0,0 +1,100 @@
+
+
+
+
diff --git a/images/milnor_singular.pdf b/images/milnor_singular.pdf
index 284c792..8769704 100644
Binary files a/images/milnor_singular.pdf and b/images/milnor_singular.pdf differ
diff --git a/images/milnor_singular.pdf_tex b/images/milnor_singular.pdf_tex
index 9fa2892..c099df0 100644
--- a/images/milnor_singular.pdf_tex
+++ b/images/milnor_singular.pdf_tex
@@ -36,7 +36,7 @@
}%
\providecommand\rotatebox[2]{#2}%
\ifx\svgwidth\undefined%
- \setlength{\unitlength}{595.27559055bp}%
+ \setlength{\unitlength}{940.6963871bp}%
\ifx\svgscale\undefined%
\relax%
\else%
@@ -48,13 +48,13 @@
\global\let\svgwidth\undefined%
\global\let\svgscale\undefined%
\makeatother%
- \begin{picture}(1,1.41428571)%
+ \begin{picture}(1,0.53281577)%
\put(0,0){\includegraphics[width=\unitlength,page=1]{milnor_singular.pdf}}%
- \put(0.65155898,1.03673474){\color[rgb]{0,0,0}\makebox(0,0)[lt]{\begin{minipage}{0.41757364\unitlength}\raggedright $F^{-1}(0)$\\ \end{minipage}}}%
- \put(0.59036283,1.14112815){\color[rgb]{0,0,0}\makebox(0,0)[lt]{\begin{minipage}{0.26998302\unitlength}\raggedright \end{minipage}}}%
- \put(0.04886191,0.32141521){\color[rgb]{0,0,0}\makebox(0,0)[lt]{\begin{minipage}{1.19914935\unitlength}\raggedright $L = F^{-1}(0) \cap S^3$\\ \end{minipage}}}%
- \put(0.63715987,1.05113382){\color[rgb]{0,0,0}\makebox(0,0)[lt]{\begin{minipage}{0.98633784\unitlength}\raggedright \end{minipage}}}%
- \put(0.86754533,0.91794223){\color[rgb]{0,0,0}\makebox(0,0)[lt]{\begin{minipage}{0.80274954\unitlength}\raggedright \end{minipage}}}%
- \put(0.86394559,0.92514175){\color[rgb]{0,0,0}\makebox(0,0)[lt]{\begin{minipage}{0.81714843\unitlength}\raggedright \end{minipage}}}%
+ \put(0.33408704,0.40878139){\color[rgb]{0,0,0}\makebox(0,0)[lt]{\begin{minipage}{0.26424189\unitlength}\raggedright $F^{-1}(0)$\\ \end{minipage}}}%
+ \put(1.33374735,-0.00561325){\color[rgb]{0,0,0}\makebox(0,0)[lt]{\begin{minipage}{0.64571752\unitlength}\raggedright \end{minipage}}}%
+ \put(0.42423363,0.20442126){\color[rgb]{0,0,0}\makebox(0,0)[lt]{\begin{minipage}{0.75882543\unitlength}\raggedright $L = F^{-1}(0) \cap S^3$\\ \end{minipage}}}%
+ \put(1.4456717,-0.22085241){\color[rgb]{0,0,0}\makebox(0,0)[lt]{\begin{minipage}{2.359021\unitlength}\raggedright \end{minipage}}}%
+ \put(1.99668384,-0.53940628){\color[rgb]{0,0,0}\makebox(0,0)[lt]{\begin{minipage}{1.91993348\unitlength}\raggedright \end{minipage}}}%
+ \put(1.98807436,-0.52218722){\color[rgb]{0,0,0}\makebox(0,0)[lt]{\begin{minipage}{1.95437124\unitlength}\raggedright \end{minipage}}}%
\end{picture}%
\endgroup%
diff --git a/images/milnor_singular.svg b/images/milnor_singular.svg
index 36417de..5c0a8fd 100644
--- a/images/milnor_singular.svg
+++ b/images/milnor_singular.svg
@@ -9,9 +9,9 @@
xmlns="http://www.w3.org/2000/svg"
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="331.85678mm"
+ height="176.81853mm"
+ viewBox="0 0 331.85678 176.81853"
version="1.1"
id="svg4463"
inkscape:version="0.92.2 5c3e80d, 2017-08-06"
@@ -26,16 +26,20 @@
inkscape:pageopacity="0.0"
inkscape:pageshadow="2"
inkscape:zoom="0.35"
- inkscape:cx="628.57143"
- inkscape:cy="-125.71429"
+ inkscape:cx="210.46118"
+ inkscape:cy="135.57601"
inkscape:document-units="mm"
inkscape:current-layer="layer1"
showgrid="false"
- inkscape:window-width="1388"
+ inkscape:window-width="1399"
inkscape:window-height="855"
- inkscape:window-x="214"
- inkscape:window-y="410"
- inkscape:window-maximized="0" />
+ inkscape:window-x="0"
+ inkscape:window-y="1"
+ inkscape:window-maximized="1"
+ fit-margin-top="0"
+ fit-margin-left="0"
+ fit-margin-right="0"
+ fit-margin-bottom="0" />
@@ -51,7 +55,8 @@
+ id="layer1"
+ transform="translate(-25.958338,-38.124052)">
@@ -85,15 +90,15 @@
id="flowPara4479" /> $L = F^{-1}(0) \cap S^3$
+ id="flowPara4525" />
diff --git a/images/torus_alpha_beta.pdf_tex b/images/torus_alpha_beta.pdf_tex
index 3a51b47..dc3d872 100644
--- a/images/torus_alpha_beta.pdf_tex
+++ b/images/torus_alpha_beta.pdf_tex
@@ -50,10 +50,10 @@
\makeatother%
\begin{picture}(1,0.41568239)%
\put(0.40129099,4.95648293){\color[rgb]{0,0,0}\makebox(0,0)[lt]{\begin{minipage}{0.37778386\unitlength}\raggedright \end{minipage}}}%
- \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(0.63089219,0.39833514){\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}}}%
+ \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}}}%
\put(0,0){\includegraphics[width=\unitlength,page=2]{torus_alpha_beta.pdf}}%
\end{picture}%
\endgroup%
diff --git a/lec_2.tex b/lec_2.tex
index 94dbb4a..1519178 100644
--- a/lec_2.tex
+++ b/lec_2.tex
@@ -277,5 +277,5 @@ H_1(M) \cong \mathbb{Z} \Longrightarrow \lambda \in \ker ( \pi_1(S^1 \times S^1)
\label{fig:meridian_and_longitude}
\end{figure}
Choose a meridian $\mu$ such that $\Lk (\mu, K) = 1$. Recall the definition of linking number via homology group (Definition \ref{def:lk_via_homo}).
-$[\mu]$ represents the generator of $H_1(S^3\setminus K, \mathbb{X})$. From definition of $\lambda$ we know that $\lambda$ is trivial in $H_1(M)$ ($\Lk(\lambda, K) =0$, therefore $[\lambda]$ was trivial in $pi_1(M)$). If $K$ is non-trivial then $\lambda$ is non-trivial in $\pi_1(M)$, but it is trivial in $H_1(M)$.
+$[\mu]$ represents the generator of $H_1(S^3\setminus K, \mathbb{Z})$. From definition of $\lambda$ we know that $\lambda$ is trivial in $H_1(M)$ ($\Lk(\lambda, K) =0$, therefore $[\lambda]$ was trivial in $pi_1(M)$). If $K$ is non-trivial then $\lambda$ is non-trivial in $\pi_1(M)$, but it is trivial in $H_1(M)$.
\end{proof}
diff --git a/lec_3.tex b/lec_3.tex
index 2aef0ee..ac2c94b 100644
--- a/lec_3.tex
+++ b/lec_3.tex
@@ -1,4 +1,4 @@
-\subsection{Algebraic knot}
+\subsection{Algebraic knots}
\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 $2$.
@@ -31,7 +31,7 @@ algebraic link (in older books on knot theory there is another notion of algebra
\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 $S^3 = \{ (z, w) \in \mathbb{C} : \max( \vert z \vert, \vert w \vert ) = \varepsilon$.
+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 $T(p, q)$.
\\???????????????????
@@ -128,9 +128,9 @@ $g_3(K_1 \# K_2) = g_3(K_1) + g_3(K_2)$.
\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.}
+\resizebox{1\textwidth}{!}{\input{images/satellite.pdf_tex}}}
+\caption{Whitehead double satellite knot.\\
+The pattern knot embedded non-trivially in an unknotted solid torus $T$ (e.i. $K \not\subset S^3\subset T$) on the left and the pattern in a companion knot - trefoil - on the right.}
\label{fig:sattelite}
\end{figure}
\noindent
diff --git a/lec_4.tex b/lec_4.tex
index 3a0a316..f2873d7 100644
--- a/lec_4.tex
+++ b/lec_4.tex
@@ -65,7 +65,7 @@ Remark: $K \sim K^{\prime} \Leftrightarrow K \# -K^{\prime}$ is slice.
\def\svgwidth{\linewidth}
\resizebox{0.5\textwidth}{!}{\input{images/ball_4_pushed_seifert.pdf_tex}}
}
-\caption{$Y = F \cup \Sigma$ is a smooth close surface.}
+\caption{$Y = F \cup \Sigma$ is a smooth closed surface.}
\label{fig:closed_surface}
\end{figure}
\noindent
diff --git a/lec_5.tex b/lec_5.tex
index 56128a2..db2b191 100644
--- a/lec_5.tex
+++ b/lec_5.tex
@@ -1,15 +1,14 @@
+\subsection{Slice knots and metabolic form}
\begin{theorem}
+\label{the:sign_slice}
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}
-\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
+If $V$ is a Hermitian matrix ($\bar{V} = V^T$), $V$ is of size $2n \times 2n$,
$
V = \begin{pmatrix}
0 & A \\
@@ -25,11 +24,16 @@ $\begin{pmatrix}
\end{pmatrix}$ 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:
+Theorem \ref{the:sign_slice} can be also express as follow:
+non-degenerate metabolic hermitian form has vanishing signature.
+\begin{proof}
+\noindent
+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 + (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)).
@@ -37,71 +41,118 @@ Let $t \in S^1 \setminus \{1\}$. Then:
As $\det (S + S^T) \neq 0$, so $S - \bar{t}S^T \neq 0$.
\end{proof}
\begin{corollary}
-If $K \sim K^\prime$ then for all but finitely many $t \in S^1 \setminus \{1\}: \sigma_K(t) = \sigma_{K^\prime}(t)$.
+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 $\mathbb{Z}$.\\
-??????????????????\\
+The signature gives 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).
+(we can use the argument that $\mathscr{C} \longrightarrow \mathbb{Z}$ as well).
\end{proof}
+\subsection{Four genus}
\begin{figure}[h]
\fontsize{20}{10}\selectfont
\centering{
\def\svgwidth{\linewidth}
-\resizebox{0.5\textwidth}{!}{\input{images/genus_2_bordism.pdf_tex}}
+\resizebox{0.7\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}
+\caption{$K$ and $K^\prime$ are connected by a genus $g$ surface.}\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 $\vert \sigma_K(t) - \sigma_{K^\prime}(t) \vert \leq 2 g$.
+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 ???
+% Borodzik 2010 Morse theory for plane algebraic curves
\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{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*}
+
+\begin{proof}
+\begin{figure}[h]
+\fontsize{20}{10}\selectfont
+\centering{
+\def\svgwidth{\linewidth}
+\resizebox{0.5\textwidth}{!}{\input{images/genus_bordism_zeros.pdf_tex}}
+}
+\caption{There exists a $3$ - manifold $\Omega$ such that $\partial \Omega = X \cup \Sigma$.}\label{fig:omega_in_B_4}
+\end{figure}
\noindent
-If $\alpha, \beta \in \ker(H_1(\Sigma \longrightarrow H_1(\Omega))$, then ${\Lk(\alpha, \beta^+) = 0}$.
+Let $K$ be a knot and $\Sigma$ its Seifert surface as in Figure \ref{fig:omega_in_B_4}.
+There exists a $3$ - submanifold
+$\Omega$ such that
+$\partial \Omega = Y = X \cup \Sigma$
+(by Thom-Pontryagin construction).
+If $\alpha, \beta \in \ker (H_1(\Sigma) \longrightarrow H_1(\Omega))$,
+then ${\Lk(\alpha, \beta^+) = 0}$. Now we have to determine the size of the kernel. We know that
+${\dim H_1(\Sigma) = 2 n}$. When we glue $\Sigma$ (genus $n$) and $X$ (genus $g$) along a circle we get a surface of genus $n + g$. Therefore $\dim H_1 (Y) = 2 n + 2 g$. Then:
+\[
+\dim (\ker (H_1(Y) \longrightarrow H_1(\Omega)) = n + g.
+\]
+So we have $H_1(W)$ of dimension
+ $2 n + 2 g$
+- the image of $H_1(Y)$
+with a subspace
+corresponding to the image of $H_1(\Sigma)$ with dimension $2 n$ and a subspace corresponding to the kernel
+of $H_1(Y) \longrightarrow H_1(\Omega)$ of size $n + g$.
+We consider minimal possible intersection of this subspaces that corresponds to the kernel of the composition $H_1(\Sigma) \longrightarrow H_1(Y) \longrightarrow H_1(\Omega)$. As the first map is injective, elements of the kernel of the composition have to be in the kernel of the second map.
+So we can calculate:
+\[
+\dim \ker (H_1(\Sigma) \longrightarrow H_1(\Omega)) = 2 n + n + g -2 n - 2 g = n - g.
+\]
+\end{proof}
\begin{corollary}
-If $t$ is nota ???? of $\det $ ????
-then $\vert \sigma_K(t) \vert \leq 2g$.\\
+If $t$ is not a root of
+$\det S S^T - $ \\
+????????????????\\
+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{fact}
+If there exists cobordism of genus $g$ between $K$ and $K^\prime$ like shown in Figure \ref{fig:proof_for_bound_disk}, then $K \# -K^\prime$ bounds a surface of genus $g$ in $B^4$.
+\end{fact}
+\begin{figure}[H]
+\fontsize{20}{10}\selectfont
+\centering{
+\def\svgwidth{\linewidth}
+\resizebox{0.7\textwidth}{!}{\input{images/genus_bordism_proof.pdf_tex}}
+}
+\caption{If $K$ and $K^\prime$ are connected by a genus $g$ surface, then $K \# -K^\prime$ bounds a genus $g$ surface.}\label{fig:proof_for_bound_disk}
+\end{figure}
+
\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.
+Remarks:
+\begin{enumerate}[label={(\arabic*)}]
+\item
+$3$ - genus is additive under taking connected sum, but $4$ - genus is not,
+\item
+for any knot $K$ we have $g_4(K) \leq g_3(K)$.
+\end{enumerate}
\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 Let $K$ be a trefoil and $K^\prime$ a mirror of a trefoil. $g_4(K^\prime) = 1$, but $g_4(K \# K^\prime) = 0$, so we see that $4$-genus isn't additive,
\item
-?????????????\\
-The equality:
+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.
+was conjecture in the '70 and proved by P. Kronheimer and T. Mrówka (1994).
+% OZSVATH-SZABO AND RASMUSSEN
\end{itemize}
\end{example}
\begin{proposition}
@@ -142,4 +193,15 @@ 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
+A form is meant as hermitian with respect to this involution: $A^T = A: (a, b) = \bar{(a, b)}$.
+\\
+????????????????????????????
+\\
+\begin{theorem}[Levine '68]
+\[
+W(\mathbb{Z}[t^{\pm 1})
+\longrightarrow \mathbb{Z}_2^\infty \oplus
+\mathbb{Z}_4^\infty \oplus
+\mathbb{Z}
+\]
+\end{theorem}
diff --git a/lec_6.tex b/lec_6.tex
index e69de29..e803a2a 100644
--- a/lec_6.tex
+++ b/lec_6.tex
@@ -0,0 +1,147 @@
+$X$ is a closed orientable four-manifold. Assume $\pi_1(X) = 0$ (it is not needed to define the intersection form). In particular $H_1(X) = 0$.
+$H_2$ is free (exercise).
+
+\begin{align*}
+H_2(X, \mathbb{Z}) \xrightarrow{\text{Poincar\'e duality}} H^2(X, \mathbb{Z} ) \xrightarrow{\text{evaluation}}\Hom(H_2(X, \mathbb{Z}), \mathbb{Z})
+\end{align*}
+
+Intersection form:
+$H_2(X, \mathbb{Z}) \times
+H_2(X, \mathbb{Z}) \longrightarrow \mathbb{Z}$ is symmetric and non singular.
+\\
+Let $A$ and $B$ be closed, oriented surfaces in $X$.
+\\
+\begin{figure}[h]
+\fontsize{20}{10}\selectfont
+\centering{
+\def\svgwidth{\linewidth}
+\resizebox{0.5\textwidth}{!}{\input{images/intersection_form_A_B.pdf_tex}}
+}
+\caption{$T_X A + T_X B = T_X X$
+}\label{fig:torus_alpha_beta}
+\end{figure}
+???????????????????????
+\begin{align*}
+x \in A \cap B\\
+T_XA \oplus T_X B = T_X X\\
+\{\epsilon_1, \dots , \epsilon_n \} = A \cap C\\
+A \cdot B = \sum^n_{i=1} \epsilon_i
+\end{align*}
+\begin{proposition}
+Intersection form $A \cdot B$ 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:
+\[
+\int_M \omega = [\omega]([M]), \quad [\omega] \in H^m_\Omega(M), \ [M] \in H_m(M).
+\]
+
+\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 $\alpha$.
+$T_X \alpha + T_X \beta = T_X \Sigma$
+}\label{fig:torus_alpha_beta}
+\end{figure}
+\begin{example}
+?????????????????????????\\
+Let $X = S^2 \times S^2$.
+We know that:
+\begin{align*}
+&H_2(S^2, \mathbb{Z}) =\mathbb{Z}\\
+&H_1(S^2, \mathbb{Z}) = 0\\
+&H_0(S^2, \mathbb{Z}) =\mathbb{Z}
+\end{align*}
+We can construct a long exact sequence for a pair:
+\begin{align*}
+&H_2(\partial X) \to H_2(X)
+\to H_2(X, \partial X) \to \\
+\to &H_1(\partial X) \to H_1(X) \to H_1(X, \partial X) \to
+\end{align*}
+????????????????????\\
+Simple case $H_1(\partial X)$ \\????????????\\
+ is torsion.
+$H_2(\partial X)$ is torsion free (by universal coefficient theorem),\\
+???????????????????????\\
+therefore it is $0$.
+\\?????????????????????\\
+We know that $b_1(X) = b_2(X)$. Therefore by Poincar\'e duality:
+\begin{align*}
+b_1(X) =
+\dim_{\mathbb{Q}} H_1(X, \mathbb{Q})
+\overset{\mathrm{PD}}{=}
+\dim_{\mathbb{Q}} H^2(X, \mathbb{Q}) =
+\dim_{\mathbb{Q}} H_2(X, \mathbb{Q}) = b_2(X)
+\end{align*}
+???????????????????????????????\\
+$H_2(X, \mathbb{Z})$ is torsion free and
+$H_2(X_1, \mathbb{Q}) = 0$, therefore $H_2(X, \mathbb{Z}) = 0$.
+The map
+$H_2(X, \mathbb{Z}) \longrightarrow H_2(X, \partial X, \mathbb{Z})$ is a monomorphism. \\??????????\\ (because it is an isomorphism after tensoring by $\mathbb{Q}$.
+\\
+Suppose $\alpha_1, \dots, \alpha_n$ is a basis of $H_2(X, \mathbb{Z})$.
+Let $A$ be the intersection matrix in this basis. Then:
+\begin{enumerate}
+\item
+A has integer coefficients,
+\item
+$\det A \neq 0$,
+\item
+$\vert \det A \vert =
+\vert H_1 (\partial X, \mathbb{Z}) \vert =
+\vert \coker H_2(X) \longrightarrow H_2(X, \partial X) \vert$.
+\end{enumerate}
+\end{example}
+???????????????????\\
+If $CVC^T = W$, then for
+$\binom{a}{b} = C^{-1} \binom{1}{0}$ we have $\binom{a}{b} $ \\
+????????????????\\
+$\omega \binom{a}{b} = \binom{1}{0} u \binom{1}{0} = 1$.
+
+\begin{theorem}[Whitehead]
+Any non-degenerate form
+\[
+A : \mathbb{Z}^4 \times \mathbb{Z}^4 \longrightarrow \mathbb{Z}
+\]
+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 definite 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 ??
+\begin{proof}
+Obviously:
+\[H_1(\partial X, \mathbb{Z}) = \coker H_2(X) \longrightarrow H_2(X, \partial X) = \quot{H_2(X, \partial X)}{H_2(X)}.
+\]
+Let $A$ be an $n \times n$ matrix. $A$ determines a \\
+??????????????/\\
+\begin{align*}
+\mathbb{Z}^n \longrightarrow \Hom (\mathbb{Z}^n, \mathbb{Z})\\
+a \mapsto (b \mapsto b^T A a)\\
+\vert \coker A \vert = \vert \det A \vert
+\end{align*}
+all homomorphisms $b = (b_1, \dots, b_n) $???????\\?????????\\
+
+\end{proof}
diff --git a/lectures_on_knot_theory.tex b/lectures_on_knot_theory.tex
index 97e49d5..13561f7 100644
--- a/lectures_on_knot_theory.tex
+++ b/lectures_on_knot_theory.tex
@@ -8,6 +8,7 @@
\usepackage[english]{babel}
+\usepackage{caption}
\usepackage{comment}
\usepackage{csquotes}
@@ -23,6 +24,7 @@
\usepackage{mathtools}
\usepackage{pict2e}
+\usepackage[section]{placeins}
\usepackage[pdf]{pstricks}
\usepackage{tikz}
@@ -84,9 +86,9 @@
\DeclareMathOperator{\Hom}{Hom}
\DeclareMathOperator{\rank}{rank}
+\DeclareMathOperator{\coker}{coker}
\DeclareMathOperator{\ord}{ord}
\DeclareMathOperator{\mytop}{top}
-
\DeclareMathOperator{\Gl}{GL}
\DeclareMathOperator{\Sl}{SL}
\DeclareMathOperator{\Lk}{lk}
@@ -126,80 +128,18 @@
%add Hurewicz theorem?
-\section{\hfill\DTMdate{2019-03-11}}
+\section{Examples of knot classes
+\hfill\DTMdate{2019-03-11}}
\input{lec_3.tex}
\section{Concordance group \hfill\DTMdate{2019-03-18}}
\input{lec_4.tex}
-
-\section{\hfill\DTMdate{2019-03-25}}
+\section{Genus $g$ cobordism \hfill\DTMdate{2019-03-25}}
\input{lec_5.tex}
\section{\hfill\DTMdate{2019-04-08}}
-%
-%
-$X$ is a closed orientable four-manifold. Assume $\pi_1(X) = 0$ (it is not needed to define the intersection form). In particular $H_1(X) = 0$.
-$H_2$ is free (exercise).
-\begin{align*}
-H_2(X, \mathbb{Z}) \xrightarrow{\text{Poincar\'e duality}} H^2(X, \mathbb{Z} ) \xrightarrow{\text{evaluation}}\Hom(H_2(X, \mathbb{Z}), \mathbb{Z})
-\end{align*}
-Intersection form:
-$H_2(X, \mathbb{Z}) \times
-H_2(X, \mathbb{Z}) \longrightarrow \mathbb{Z}$ - 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:
-\[
-[A], [B] \in H_2(X, \mathbb{Z}).
-\]
-\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:
-\[
-\int_M \omega = [\omega]([M]), \quad [\omega] \in H^m_\Omega(M), \ [M] \in H_m(M).
-\]
-
-\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 ??
-
+\input{lec_6.tex}
\section{\hfill\DTMdate{2019-04-15}}
\begin{theorem}
@@ -341,14 +281,17 @@ a &\mapsto (a, \_) H_2(M, \mathbb{Z})
\end{align*}
has coker precisely $H_1(Y, \mathbb{Z})$.
\\???????????????\\
-Let $K \subset S^3$ be a knot, \\
-$X = S^3 \setminus K$ - a knot complement, \\
-$\widetilde{X} \xrightarrow{\enspace \rho \enspace} X$ - an infinite cyclic cover (universal abelian cover).
+Let $K \subset S^3$ be a knot, $X = S^3 \setminus K$ a knot complement and
+$\widetilde{X} \xrightarrow{\enspace \rho \enspace} X$ an infinite cyclic cover (universal abelian cover). By Hurewicz theorem we know that:
\begin{align*}
\pi_1(X) \longrightarrow \quot{\pi_1(X)}{[\pi_1(X), \pi_1(X)]} = H_1(X, \mathbb{Z} ) \cong \mathbb{Z}
\end{align*}
+????????????????????????????????????????????????????????????????????????\\
+????????????????????????????????????????????????????????????????????????\\
+????????????????????????????????????????????????????????????????????????\\
+????????????????????????????????????????????????????????????????????????\\
$C_{*}(\widetilde{X})$ has a structure of a $\mathbb{Z}[t, t^{-1}] \cong \mathbb{Z}[\mathbb{Z}]$ module. \\
-$H_1(\widetilde{X}, \mathbb{Z}[t, t^{-1}])$ - Alexander module, \\
+Let $H_1(\widetilde{X}, \mathbb{Z}[t, t^{-1}])$ be the Alexander module of the knot $K$ with an intersection form:
\begin{align*}
H_1(\widetilde{X}, \mathbb{Z}[t, t^{-1}]) \times
H_1(\widetilde{X}, \mathbb{Z}[t, t^{-1}]) \longrightarrow \quot{\mathbb{Q}}{\mathbb{Z}[t, t^{-1}]}
@@ -365,27 +308,31 @@ H_1(\widetilde{X}, \mathbb{Z}[t, t^{-1}]) \longrightarrow \quot{\mathbb{Q}}{\mat
\begin{align*}
H_1(\widetilde{X}, \mathbb{Z}[t, t^{-1}]) \times
H_1(\widetilde{X}, \mathbb{Z}[t, t^{-1}]) &\longrightarrow \quot{\mathbb{Q}}{\mathbb{Z}[t, t^{-1}]}\\
-(\alpha, \beta) &\mapsto \alpha^{-1}(t -1)(tV - V^T)^{-1}\beta
+(\alpha, \beta) \quad &\mapsto \alpha^{-1}(t -1)(tV - V^T)^{-1}\beta
\end{align*}
\end{fact}
\noindent
-Note that $\mathbb{Z}$ is not PID. Therefore we don't have primer decomposition of this moduli. We can simplify this problem by replacing $\mathbb{Z}$ by $\mathbb{R}$. We lose some date by doing this transition.
+Note that $\mathbb{Z}$ is not PID.
+Therefore we don't have primary decomposition of this module.
+We can simplify this problem by replacing $\mathbb{Z}$ by $\mathbb{R}$. We lose some date by doing this transition, but we can
\begin{align*}
-&\xi \in S^1 \setminus \{ \pm 1\}
-\quad
+\xi \in S^1 \setminus \{ \pm 1\}
+&\quad
p_{\xi} =
(t - \xi)(t - \xi^{-1}) t^{-1}
\\
-&\xi \in \mathbb{R} \setminus \{ \pm 1\}
-\quad
+\xi \in \mathbb{R} \setminus \{ \pm 1\}
+&\quad
q_{\xi} = (t - \xi)(t - \xi^{-1}) t^{-1}
\\
-&
-\xi \notin \mathbb{R} \cup S^1 \quad
-q_{\xi} = (t - \xi)(t - \overbar{\xi})(t - \xi^{-1})(t - \overbar{\xi}^{-1}) t^{-2}\\
-&
-\Lambda = \mathbb{R}[t, t^{-1}]\\
-&\text{Then: } H_1(\widetilde{X}, \Lambda) \cong \bigoplus_{\substack{\xi \in S^1 \setminus \{\pm 1 \}\\ k\geq 0}}
+\xi \notin \mathbb{R} \cup S^1
+&\quad
+q_{\xi} = (t - \xi)(t - \overbar{\xi})(t - \xi^{-1})
+(t - \overbar{\xi}^{-1}) t^{-2}
+\end{align*}
+Let $\Lambda = \mathbb{R}[t, t^{-1}]$. Then:
+\begin{align*}
+H_1(\widetilde{X}, \Lambda) \cong \bigoplus_{\substack{\xi \in S^1 \setminus \{\pm 1 \}\\ k\geq 0}}
( \quot{\Lambda}{p_{\xi}^k })^{n_k, \xi}
\oplus
\bigoplus_{\substack{\xi \notin S^1 \\ l\geq 0}}
@@ -557,7 +504,9 @@ $2 \sum\limits_{k_i \text{ odd}} \epsilon_i$. The peak of the signature function
....
\begin{definition}
-A square hermitian matrix $A$ of size $n$.
+A square hermitian matrix $A$ of size $n$ with coefficients in \\
+the Blanchfield pairing if:
+$H_1(\bar{X}$
\end{definition}
field of fractions