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deleted file mode 100644
index 0858b16..0000000
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deleted file mode 100644
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new file mode 100644
index 0000000..5a9c752
--- /dev/null
+++ b/images/linking_hopf.pdf_tex
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+%% Creator: Inkscape inkscape 0.91, www.inkscape.org
+%% PDF/EPS/PS + LaTeX output extension by Johan Engelen, 2010
+%% Accompanies image file 'linking_hopf.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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new file mode 100644
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+%% Creator: Inkscape inkscape 0.91, www.inkscape.org
+%% PDF/EPS/PS + LaTeX output extension by Johan Engelen, 2010
+%% Accompanies image file 'linking_torus_6_2.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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new file mode 100644
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+%% Creator: Inkscape inkscape 0.91, www.inkscape.org
+%% PDF/EPS/PS + LaTeX output extension by Johan Engelen, 2010
+%% Accompanies image file 'seifert_alg.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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new file mode 100644
index 0000000..6416c24
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diff --git a/images/seifert_matrix.pdf_tex b/images/seifert_matrix.pdf_tex
new file mode 100644
index 0000000..4851a84
--- /dev/null
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+%% Creator: Inkscape inkscape 0.91, www.inkscape.org
+%% PDF/EPS/PS + LaTeX output extension by Johan Engelen, 2010
+%% Accompanies image file 'seifert_matrix.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(0,0){\includegraphics[width=\unitlength,page=1]{seifert_matrix.pdf}}%
+ \put(0.62239545,0.32756023){\color[rgb]{0,0,0}\makebox(0,0)[lt]{\begin{minipage}{0.15393011\unitlength}\raggedright $\alpha_1^+$\end{minipage}}}%
+ \put(0.69294518,0.43731309){\color[rgb]{0,0,0}\makebox(0,0)[lt]{\begin{minipage}{0.14012473\unitlength}\raggedright \end{minipage}}}%
+ \put(0.68037809,0.26129435){\color[rgb]{0,0,0}\makebox(0,0)[lt]{\begin{minipage}{0.12493878\unitlength}\raggedright $\alpha_1$\end{minipage}}}%
+ \put(0,0){\includegraphics[width=\unitlength,page=2]{seifert_matrix.pdf}}%
+ \put(0.13859342,0.26031816){\color[rgb]{0,0,0}\makebox(0,0)[lt]{\begin{minipage}{0.12493878\unitlength}\raggedright $\alpha_1$\end{minipage}}}%
+ \put(0.15714098,0.12950888){\color[rgb]{0,0,0}\makebox(0,0)[lt]{\begin{minipage}{0.12493878\unitlength}\raggedright $\alpha_2$\end{minipage}}}%
+ \put(0,0){\includegraphics[width=\unitlength,page=3]{seifert_matrix.pdf}}%
+ \end{picture}%
+\endgroup%
diff --git a/images/seifert_matrix.svg b/images/seifert_matrix.svg
new file mode 100644
index 0000000..ee448a3
--- /dev/null
+++ b/images/seifert_matrix.svg
@@ -0,0 +1,2393 @@
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diff --git a/images/seifert_surface.svg b/images/seifert_surface.svg
index b330211..19770fd 100644
--- a/images/seifert_surface.svg
+++ b/images/seifert_surface.svg
@@ -23,6 +23,145 @@
inkscape:export-ydpi="90">
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diff --git a/images/seifert_surface.svg.2019_06_02_17_48_53.0.svg b/images/seifert_surface.svg.2019_06_02_17_48_53.0.svg
new file mode 100644
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diff --git a/images/seifert_bands.svg b/images/seifert_surface_v2.svg
similarity index 63%
rename from images/seifert_bands.svg
rename to images/seifert_surface_v2.svg
index c79e5b4..23fd6c8 100644
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+ viewBox="0 0 1280.5281 1342.0375"
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inkscape:version="0.91 r13725"
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+ sodipodi:docname="seifert_surface_v2.svg"
+ inkscape:export-filename="/Users/Kasia/lectures_on_knot_theory/images/seifert3d.png"
+ inkscape:export-xdpi="90"
+ inkscape:export-ydpi="90">
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$B^4$
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diff --git a/images/seifert_surface_v2.svg.2019_06_02_17_48_53.1.svg b/images/seifert_surface_v2.svg.2019_06_02_17_48_53.1.svg
new file mode 100644
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diff --git a/images/torus_1_2_3.pdf b/images/torus_1_2_3.pdf
new file mode 100644
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diff --git a/images/torus_1_2_3.pdf_tex b/images/torus_1_2_3.pdf_tex
new file mode 100644
index 0000000..b882928
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+%% Accompanies image file 'torus_1_2_3.pdf' (pdf, eps, ps)
+%%
+%% To include the image in your LaTeX document, write
+%% \input{.pdf_tex}
+%% instead of
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+%% To scale the image, write
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+%% \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
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+%% in the preamble, and then including the image with
+%% \import{}{.pdf_tex}
+%% Alternatively, one can specify
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+%%
+%% For more information, please see info/svg-inkscape on CTAN:
+%% http://tug.ctan.org/tex-archive/info/svg-inkscape
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+ \put(0.87444349,0.06347455){\color[rgb]{0,0,0}\makebox(0,0)[lt]{\begin{minipage}{0.1438923\unitlength}\raggedright genus $3$\\ \end{minipage}}}%
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+\endgroup%
diff --git a/images/torus_1_2_3.svg b/images/torus_1_2_3.svg
new file mode 100644
index 0000000..41561e2
--- /dev/null
+++ b/images/torus_1_2_3.svg
@@ -0,0 +1,2101 @@
+
+
+
+
diff --git a/lectures_on_knot_theory.tex b/lectures_on_knot_theory.tex
index 746362f..99b6955 100644
--- a/lectures_on_knot_theory.tex
+++ b/lectures_on_knot_theory.tex
@@ -48,34 +48,41 @@
{\bfseries}{}%
{\newline}{}%
\theoremstyle{break}
-\newtheorem{lemma}{Lemma}
-\newtheorem{fact}{Fact}
-\newtheorem{corollary}{Corollary}
-\newtheorem{example}{Example}
-\newtheorem{definition}{Definition}
-\newtheorem{theorem}{Theorem}
-\newtheorem{proposition}{Proposition}
-
+\newtheorem{lemma}{Lemma}[section]
+\newtheorem{fact}{Fact}[section]
+\newtheorem{corollary}{Corollary}[section]
+\newtheorem{proposition}{Proposition}[section]
+\newtheorem{example}{Example}[section]
+\newtheorem{definition}{Definition}[section]
+\newtheorem{theorem}{Theorem}[section]
\newcommand{\contradiction}{%
\ensuremath{{\Rightarrow\mspace{-2mu}\Leftarrow}}}
\newcommand*\quot[2]{{^{\textstyle #1}\big/_{\textstyle #2}}}
+
\newcommand{\overbar}[1]{%
\mkern 1.5mu=\overline{%
\mkern-1.5mu#1\mkern-1.5mu}%
\mkern 1.5mu}
+\newcommand{\sdots}{\smash{\vdots}}
+
\AtBeginDocument{\renewcommand{\setminus}{%
\mathbin{\backslash}}}
\DeclareMathOperator{\Hom}{Hom}
\DeclareMathOperator{\rank}{rank}
+\DeclareMathOperator{\Gl}{Gl}
+
+\titleformat{\section}{\normalfont \fontsize{12}{15} \bfseries}{%
+ Lecture\ \thesection}%
+ {2.3ex plus .2ex}{}
+\titlespacing*{\section}
+{0pt}{16.5ex plus 1ex minus .2ex}{4.3ex plus .2ex}
-\titleformat{\section}{\normalfont \large \bfseries}{%
- Lecture\ \thesection}{2.3ex plus .2ex}{}
-
-%\setlist[itemize]{topsep=0pt,before=%\leavevmode\vspace{0.5em}}
+\setlist[itemize]{topsep=0pt,before=%
+\leavevmode\vspace{0.5em}}
\input{knots_macros}
@@ -87,7 +94,7 @@
%\newpage
%\input{myNotes}
-\section{\hfill\DTMdate{2019-02-25}}
+\section{Basic definitions \hfill\DTMdate{2019-02-25}}
\begin{definition}
A knot $K$ in $S^3$ is a smooth (PL - smooth) embedding of a circle $S^1$ in $S^3$:
\begin{align*}
@@ -96,12 +103,13 @@ A knot $K$ in $S^3$ is a smooth (PL - smooth) embedding of a circle $S^1$ in $S^
\end{definition}
\noindent
Usually we think about a knot as an image of an embedding: $K = \varphi(S^1)$.
+
\begin{example}
\begin{itemize}
\item
Knots:
-\includegraphics[width=0.08\textwidth]{unknot.png},
-\includegraphics[width=0.08\textwidth]{trefoil.png}.
+\includegraphics[width=0.08\textwidth]{unknot.png} (unknot),
+\includegraphics[width=0.08\textwidth]{trefoil.png} (trefoil).
\item
Not knots:
\includegraphics[width=0.12\textwidth]{not_injective_knot.png}
@@ -172,7 +180,7 @@ Let $D$ be a diagram of an oriented link (to each component of a link we add an
We can distinguish two types of crossings: right-handed
$\left(\PICorientpluscross\right)$, called a positive crossing, and left-handed $\left(\PICorientminuscross\right)$, called a negative crossing.
-\section*{Reidemeister moves}
+\subsection{Reidemeister moves}
A Reidemeister move is one of the three types of operation on a link diagram as shown below:
\begin{enumerate}[label=\Roman*]
\item\hfill\\
@@ -197,7 +205,7 @@ deformed into each other by a finite sequence of Reidemeister moves (and isotopy
%Singularities of Differentiable Maps
%Authors: Arnold, V.I., Varchenko, Alexander, Gusein-Zade, S.M.
-\subsection*{Seifert surface}
+\subsection{Seifert surface}
\noindent
Let $D$ be an oriented diagram of a link $L$. We change the diagram by smoothing each crossing:
\begin{align*}
@@ -221,7 +229,7 @@ Note: in general the obtained surface doesn't need to be connected, but by takin
\begin{figure}[H]
\begin{center}
-\includegraphics[width=0.6\textwidth]{seifert_connect.png}
+\includegraphics[width=0.4\textwidth]{seifert_connect.png}
\end{center}
\caption{Connecting two surfaces.}
\label{fig:SeifertConnect}
@@ -259,13 +267,16 @@ On a diagram $L$ consider all crossings between $\alpha$ and $\beta$. Let $N_+$
\end{definition}
\hfill
\\
-Let $\nu(\beta)$ be a tubular neighbourhood of a closed simple curve $\beta$. The linking number can be interpreted via first homology group, where $lk(\alpha, \beta)$ is equal to evaluation of $\alpha$ as element of first homology group in complement of $\beta$ in $S^3$:
+Let $\alpha$ and $\beta$ be two disjoint simple cross curves in $S^3$.
+Let $\nu(\beta)$ be a tubular neighbourhood of $\beta$. The linking number can be interpreted via first homology group, where $lk(\alpha, \beta)$ is equal to evaluation of $\alpha$ as element of first homology group of the complement of $\beta$:
\[
\alpha \in H_1(S^3 \setminus \nu(\beta), \mathbb{Z}) \cong \mathbb{Z}.\]
+
+
\begin{example}
-\begin{itemize}\hfill
+\begin{itemize}
\item
-Hopf link\hfill
+Hopf link
\begin{figure}[H]
\fontsize{20}{10}\selectfont
\centering{
@@ -274,7 +285,7 @@ Hopf link\hfill
}
\end{figure}
\item
-$T(6, 2)$ link\hfill
+$T(6, 2)$ link
\begin{figure}[H]
\fontsize{20}{10}\selectfont
\centering{
@@ -285,94 +296,263 @@ $T(6, 2)$ link\hfill
\end{itemize}
\end{example}
-Let $L$ be a link and $\Sigma$ be a Seifert surface for $L$. Choose a basis for $H_1(\Sigma, \mathbb{Z})$ consisting of simple closed $\alpha_1, \dots, \alpha_n$.
-Let $\alpha_1^+, \dots \alpha_n^+$ be copies of $\alpha_i$ lifted up off the surface. Let $lk(\alpha_i, \alpha_i^+) = \{a_{ij}\}$. Then the matrix $S = \{a_{ij}\}_{i, j =1}^n$ is called a Seifert matrix for $L$.
+\subsection{Seifert matrix}
+Let $L$ be a link and $\Sigma$ be an oriented Seifert surface for $L$. Choose a basis for $H_1(\Sigma, \mathbb{Z})$ consisting of simple closed $\alpha_1, \dots, \alpha_n$.
+Let $\alpha_1^+, \dots \alpha_n^+$ be copies of $\alpha_i$ lifted up off the surface (push up along a vector field normal to $\Sigma$). Note that elements $\alpha_i$ are contained in the Seifert surface while all $\alpha_i^+$ are don't intersect the surface.
+Let $lk(\alpha_i, \alpha_j^+) = \{a_{ij}\}$. Then the matrix $S = \{a_{ij}\}_{i, j =1}^n$ is called a Seifert matrix for $L$. Note that by choosing a different basis we get a different matrix.
+
+\begin{figure}[H]
+\fontsize{20}{10}\selectfont
+\centering{
+\def\svgwidth{\linewidth}
+\resizebox{0.8\textwidth}{!}{\input{images/seifert_matrix.pdf_tex}}
+}
+\end{figure}
\begin{theorem}
The Seifert matrices $S_1$ and $S_2$ for the same link $L$ are S-equivalent, that is, $S_2$ can be obtained from $S_1$ by a sequence of following moves:
\begin{enumerate}[label={(\arabic*)}]
+
\item
-$V \rightarrow AVA^T$ for $A \in $
+$V \rightarrow AVA^T$, where $A$ is a matrix with integer coefficients,
+
\item
+
$V \rightarrow
\begin{pmatrix}
- \alpha & * \\
- \gamma^{*} & \delta
-\end{pmatrix}
-$\\
-\[
- \begin{pmatrix}
\begin{array}{c|c}
- \epsilon' [T|_A]\epsilon & \ast \\
- \hline
- 0 & _{\overline{B}'} [\overline{T}]
- _{\overline{B}\vphantom{\overline{B}'}}
+ V &
+ \begin{matrix}
+ \ast & 0 \\
+ \sdots & \sdots\\
+ \ast & 0
+ \end{matrix} \\
+ \hline
+ \begin{matrix}
+ \ast & \dots & \ast\\
+ 0 & \dots & 0
+ \end{matrix}
+ &
+ \begin{matrix}
+ 0 & 0\\
+ 1 & 0
+ \end{matrix}
\end{array}
- \end{pmatrix}
-\]\\
-\[\left|
-\begin{array}{cr}
- Q & \begin{matrix} 0 \\ 0 \end{matrix} \\
- \begin{matrix} 2 & 3 \end{matrix} & -1
-\end{array}
-\right|\]
-\\
-\[
-\left[
-\begin{array}{c@{}c@{}c}
- \left[\begin{array}{cc}
- a_{11} & a_{12} \\
- a_{21} & a_{22} \\
- \end{array}\right] & \mathbf{0} & \mathbf{0} \\
- \mathbf{0} & \left[\begin{array}{ccc}
- b_{11} & b_{12} & b_{13}\\
- b_{21} & b_{22} & b_{23}\\
- b_{31} & b_{32} & b_{33}\\
- \end{array}\right] & \mathbf{0}\\
-\mathbf{0} & \mathbf{0} & \left[ \begin{array}{cc}
-c_{11} & c_{12} \\
-c_{21} & c_{22} \\
-\end{array}\right] \\
-\end{array}\right]
-\] \\
-\[
-\begin{bmatrix}
- \begin{bmatrix}
- a_{11} & a_{12}\\
- a_{21} & a_{22}\\
- \end{bmatrix} & \mathbf{0} & \mathbf{0} \\
- \mathbf{0} & \begin{bmatrix}
- b_{11} & b_{12} & b_{13}\\
- b_{21} & b_{22} & b_{23}\\
- b_{31} & b_{32} & b_{33}\\
- \end{bmatrix} & \mathbf{0} \\
- \mathbf{0} & \mathbf{0} & \begin{bmatrix}
- c_{11} & c_{12}\\
- c_{21} & c_{22}\\
- \end{bmatrix} \\
-\end{bmatrix}
-\]\\
-\setlength{\arraycolsep}{2em}
-\newcommand{\lbrce}{\smash{\left.\rule{0pt}{25pt}\right\}}}
-\newcommand{\rbrce}{\smash{\left\{\rule{0pt}{25pt}\right.}}
-\newcommand{\sdots}{\smash{\vdots}}
-\[
- \begin{pmatrix}
- 0 & 0 & 0 \\
- \sdots & \sdots\makebox[0pt][l]{$\lbrce\left\lceil\frac i2\right\rceil$} & \sdots \\
- 0 & 0 & \\
- & & 0 \\
- & & \makebox[0pt][r]{$\left\lfloor\frac i2\right\rfloor\rbrce$}\sdots \\
- 0 & & 0
- \end{pmatrix}
-\]
-
+\end{pmatrix} \quad$
+or
+$\quad
+V \rightarrow
+\begin{pmatrix}
+ \begin{array}{c|c}
+ V &
+ \begin{matrix}
+ \ast & 0 \\
+ \sdots & \sdots\\
+ \ast & 0
+ \end{matrix} \\
+ \hline
+ \begin{matrix}
+ \ast & \dots & \ast\\
+ 0 & \dots & 0
+ \end{matrix}
+ &
+ \begin{matrix}
+ 0 & 1\\
+ 0 & 0
+ \end{matrix}
+ \end{array}
+\end{pmatrix}$
\item
inverse of (2)
\end{enumerate}
\end{theorem}
+\section{\hfill\DTMdate{2019-03-04}}
+\begin{theorem}
+For any knot $K \subset S^3$ there exists a connected, compact and orientable surface $\Sigma(K)$ such that $\partial \Sigma(K) = K$
+\end{theorem}
+\begin{proof}("joke")\\
+Let $K \in S^3$ be a knot and $N = \nu(K)$ be its tubular neighbourhood. Because $K$ and $N$ are homotopy equivalent, we get:
+\begin{align*}
+H^1(S^3 \setminus N ) \cong H^1(S^3 \setminus K).
+\end{align*}
+Let us consider a long exact sequence of cohomology of a pair $(S^3, S^3 \setminus N)$ with integer coefficients:
+
+\begin{center}
+\begin{tikzcd}
+[
+ column sep=0cm, fill=none,
+ row sep=small,
+ ar symbol/.style =%
+ {draw=none,"\textstyle#1" description,sloped},
+ isomorphic/.style = {ar symbol={\cong}},
+]
+&\mathbb{Z}
+\\
+
+& H^0(S^3) \ar[u,isomorphic] \to
+&H^0(S^3 \setminus N) \to
+\\
+\to H^1(S^3, S^3 \setminus N) \to
+ & H^1(S^3) \to
+ & H^1(S^3\setminus N) \to
+ \\
+& 0 \ar[u,isomorphic]&
+ \\
+ \to H^2(S^3, S^3 \setminus N) \to
+ & H^2(S^3) \ar[u,isomorphic] \to
+ & H^2(S^3\setminus N) \to
+ \\
+\to H^3(S^3, S^3\setminus N)\to
+& H^3(S) \to
+& 0
+\\
+& \mathbb{Z} \ar[u,isomorphic] &\\
+ \end{tikzcd}
+\end{center}
+\[
+H^* (S^3, S^3 \setminus N) \cong H^* (N, \partial N)
+\]
+\\
+??????????????
+\\
+
+\end{proof}
+
+\begin{definition}
+Let $S$ be a Seifert matrix for a knot $K$. The Alexander polynomial $\Delta_K(t)$ is a Laurent polynomial:
+\[
+\Delta_K(t) := \det (tS - S^T) \in
+\mathbb{Z}[t, t^{-1}] \cong \mathbb{Z}[\mathbb{Z}]
+\]
+\end{definition}
+
+\begin{theorem}
+$\Delta_K(t)$ is well defined up to multiplication by $\pm t^k$, for $k \in \mathbb{Z}$.
+\end{theorem}
+\begin{proof}
+We need to show that $\Delta_K(t)$ doesn't depend on $S$-equivalence relation.
+\begin{enumerate}[label={(\arabic*)}]
+\item Suppose $S\prime = CSC^T$, $C \in \Gl(n, \mathbb{Z})$ (matrices invertible over $\mathbb{Z}$). Then $\det C = 1$ and:
+\begin{align*}
+&\det(tS\prime - S\prime^T) =
+\det(tCSC^T - (CSC^T)^T) =\\
+&\det(tCSC^T - CS^TC^T) =
+\det C(tS - S^T)C^T =
+\det(tS - S^T)
+\end{align*}
+\item
+Let \\
+$ A := t
+\begin{pmatrix}
+ \begin{array}{c|c}
+ S &
+ \begin{matrix}
+ \ast & 0 \\
+ \sdots & \sdots\\
+ \ast & 0
+ \end{matrix} \\
+ \hline
+ \begin{matrix}
+ \ast & \dots & \ast\\
+ 0 & \dots & 0
+ \end{matrix}
+ &
+ \begin{matrix}
+ 0 & 0\\
+ 1 & 0
+ \end{matrix}
+ \end{array}
+\end{pmatrix}
+-
+\begin{pmatrix}
+ \begin{array}{c|c}
+ S^T &
+ \begin{matrix}
+ \ast & 0 \\
+ \sdots & \sdots\\
+ \ast & 0
+ \end{matrix} \\
+ \hline
+ \begin{matrix}
+ \ast & \dots & \ast\\
+ 0 & \dots & 0
+ \end{matrix}
+ &
+ \begin{matrix}
+ 0 & 1\\
+ 0 & 0
+ \end{matrix}
+ \end{array}
+\end{pmatrix}
+=
+\begin{pmatrix}
+ \begin{array}{c|c}
+ tS - S^T &
+ \begin{matrix}
+ \ast & 0 \\
+ \sdots & \sdots\\
+ \ast & 0
+ \end{matrix} \\
+ \hline
+ \begin{matrix}
+ \ast & \dots & \ast\\
+ 0 & \dots & 0
+ \end{matrix}
+ &
+ \begin{matrix}
+ 0 & -1\\
+ t & 0
+ \end{matrix}
+ \end{array}
+\end{pmatrix}
+$
+\\
+\\
+Using the Laplace expansion we get $\det A = \pm t \det(tS - S^T)$.
+\end{enumerate}
+\end{proof}
+%
+%
+%
+\begin{example}
+If $K$ is a trefoil then we can take
+$S = \begin{pmatrix}
+-1 & -1 \\
+0 & -1
+\end{pmatrix}$.
+\[
+\Delta_K(t) = \det
+\begin{pmatrix}
+-t + 1 & -t\\
+1 & -t +1
+\end{pmatrix}
+= (t -1)^2 + t = t^2 - t +1 \ne 1
+\Rightarrow \text{trefoil is not trivial}
+\]
+\end{example}
+\begin{fact}
+$\Delta_K(t)$ is symmetric.
+\end{fact}
+\begin{proof}
+Let $S$ be an $n \times n$ matrix.
+\begin{align*}
+&\Delta_K(t^{-1}) = \det (t^{-1}S - S^T) = (-t)^{-n} \det(tS^T - S) = \\
+&(-t)^{-n} \det (tS - S^T) = (-t)^{-n} \Delta_K(t)
+\end{align*}
+If $K$ is a knot, then $n$ is necessarily even, and so $\Delta_K(t^{-1}) = t^{-n} \Delta_K(t)$.
+\end{proof}
+\begin{lemma}
+\begin{align*}
+\frac{1}{2} \deg \Delta_K(t) \leq g_3(K),
+\text{ where } deg (a_n t^n + \cdots + a_1 t^l )= k - l.
+\end{align*}
+\end{lemma}
+\begin{proof}
+
+\end{proof}
+%removing one disk from surface doesn't change $H_1$ (only $H_2$)
\section{}
\begin{example}
\begin{align*}
@@ -389,17 +569,6 @@ Prove that if $K$ is negative amphichiral, then $K \# K$ in
$\mathbf{C}$
\end{example}
-\section{\hfill\DTMdate{2019-03-04}}
-\begin{proof}("joke")\\
-Let $K \in S^3$ be a knot and $N$ be its tubular neighbourhood.
-\begin{align*}
-H^1(S^3 \setminus N ) \cong H^1(S^3 \setminus K)
-\end{align*}
-For a pair $(S^3, S^3 \setminus N)$ we have:
-\begin{align*}
-H^0(S^3)
-\end{align*}
-\end{proof}
\section{\hfill\DTMdate{2019-03-18}}
\begin{definition}
A knot $K$ is called (smoothly) slice if $K$ is smoothly concordant to an unknot. \\
@@ -447,7 +616,11 @@ $H_2(X, \mathbb{Z}) \times
H_2(X, \mathbb{Z}) \longrightarrow \mathbb{Z}$ - 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.
+%$A \cdot B$ gives the pairing as ??
+
+\end{proposition}
\section{\hfill\DTMdate{2019-04-15}}
In other words:\\
@@ -459,27 +632,28 @@ of $H_2(Y, \mathbb{Z}$, then $A = (b_i, b_y)$ \\??\\ is a matrix of intersection
\end{align*}
In particular $\mid \det A\mid = \# H_1(Y, \mathbb{Z}$.\\
That means - what is happening on boundary is a measure of degeneracy.
-\\
-\vspace{1cm}
+
\begin{center}
\begin{tikzcd}
[
-column sep=tiny,
-row sep=small,
-ar symbol/.style = {draw=none,"\textstyle#1" description,sloped},
-isomorphic/.style = {ar symbol={\cong}},
+ column sep=tiny,
+ row sep=small,
+ ar symbol/.style =%
+ {draw=none,"\textstyle#1" description,sloped},
+ isomorphic/.style = {ar symbol={\cong}},
]
-H_1(Y, \mathbb{Z})&
-\times \quad H_1(Y, \mathbb{Z})&
-\longrightarrow &
-\quot{\mathbb{Q}}{\mathbb{Z}}
- \text{ - a linking form}
+H_1(Y, \mathbb{Z}) &
+ \times \quad H_1(Y, \mathbb{Z})&
+ \longrightarrow &
+ \quot{\mathbb{Q}}{\mathbb{Z}}
+ \text{ - a linking form}
\\
-\quot{\mathbb{Z}^n}{A\mathbb{Z}} \ar[u,isomorphic] &
-\quot{\mathbb{Z}^n}{A\mathbb{Z}} \ar[u,isomorphic] &\\
+ \quot{\mathbb{Z}^n}{A\mathbb{Z}} \ar[u,isomorphic] &
+ \quot{\mathbb{Z}^n}{A\mathbb{Z}} \ar[u,isomorphic] &\\
\end{tikzcd}
$(a, b) \mapsto aA^{-1}b^T$
\end{center}
+
The intersection form on a four-manifold determines the linking on the boundary. \\
\noindent
@@ -780,4 +954,22 @@ field of fractions
\section{balagan}
+\begin{comment}
+\setlength{\arraycolsep}{2em}
+\newcommand{\lbrce}{\smash{\left.\rule{0pt}{25pt}\right\}}}
+\newcommand{\rbrce}{\smash{\left\{\rule{0pt}{25pt}\right.}}
+\[
+ \begin{pmatrix}
+ 0 & 0 & 0 \\
+ \sdots & \sdots\makebox[0pt][l]{$\lbrce\left\lceil\frac i2\right\rceil$} & \sdots \\
+ 0 & 0 & \\
+ \hline
+
+ & & 0 \\
+ & & \makebox[0pt][r]{$\left\lfloor\frac i2\right\rfloor\rbrce$}\sdots \\
+ 0 & & 0
+ \end{pmatrix}
+\]
+
+\end{comment}
\end{document}