matrice
This commit is contained in:
Maria Marchwicka
parent
11a97cecc9
commit
d2146a556c
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|
||||
id="flowPara18462">\shortstack{$lk(\alpha, \beta) = 3$}</flowPara></flowRoot> <flowRoot
|
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style="font-style:normal;font-variant:normal;font-weight:normal;font-stretch:normal;font-size:11.9998827px;line-height:125%;font-family:sans-serif;-inkscape-font-specification:'sans-serif, Normal';font-variant-ligatures:normal;font-variant-caps:normal;font-variant-numeric:normal;font-feature-settings:normal;text-align:start;writing-mode:lr-tb;text-anchor:start" /></flowRegion><flowPara
|
||||
id="flowPara3435">\shortstack{$\Lk(\alpha, \beta) = 3$}</flowPara></flowRoot> <flowRoot
|
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transform="matrix(0.39968512,0,0,0.29332933,182.9764,926.96522)"
|
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style="font-style:normal;font-variant:normal;font-weight:normal;font-stretch:normal;font-size:40.00024796px;line-height:125%;font-family:sans-serif;-inkscape-font-specification:'sans-serif, Normal';text-align:start;letter-spacing:0px;word-spacing:0px;writing-mode:lr-tb;text-anchor:start;fill:#ff0000;fill-opacity:1;stroke:none;stroke-width:1px;stroke-linecap:butt;stroke-linejoin:miter;stroke-opacity:1"
|
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style="font-style:normal;font-variant:normal;font-weight:normal;font-stretch:normal;line-height:0.01%;font-family:sans-serif;-inkscape-font-specification:'sans-serif, Normal';text-align:start;letter-spacing:0px;word-spacing:0px;writing-mode:lr-tb;text-anchor:start;fill:#ff0000;fill-opacity:1;stroke:none;stroke-width:1px;stroke-linecap:butt;stroke-linejoin:miter;stroke-opacity:1"
|
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id="flowRoot11275"
|
||||
xml:space="preserve"><flowRegion
|
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id="flowRegion11277"><rect
|
||||
|
|
@ -2064,11 +1943,13 @@
|
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height="78.152328"
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width="237.64688"
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id="rect11279" /></flowRegion><flowPara
|
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id="flowPara11285">$\alpha$</flowPara><flowPara
|
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id="flowPara11287"></flowPara></flowRoot> <flowRoot
|
||||
id="flowPara11285"
|
||||
style="font-size:40.00024796px;line-height:1.25">$\alpha$</flowPara><flowPara
|
||||
id="flowPara11287"
|
||||
style="font-size:40.00024796px;line-height:1.25"> </flowPara></flowRoot> <flowRoot
|
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xml:space="preserve"
|
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id="flowRoot11289"
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style="font-style:normal;font-variant:normal;font-weight:normal;font-stretch:normal;line-height:0.01%;font-family:sans-serif;-inkscape-font-specification:'sans-serif, Normal';text-align:start;letter-spacing:0px;word-spacing:0px;writing-mode:lr-tb;text-anchor:start;fill:#0000ff;fill-opacity:1;stroke:none;stroke-width:1px;stroke-linecap:butt;stroke-linejoin:miter;stroke-opacity:1"
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|
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id="flowRegion11291"><rect
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|
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|
|
@ -2077,6 +1958,8 @@
|
|||
x="548.66125"
|
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y="252.68971"
|
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style="font-style:normal;font-variant:normal;font-weight:normal;font-stretch:normal;font-size:40.00024796px;line-height:125%;font-family:sans-serif;-inkscape-font-specification:'sans-serif, Normal';text-align:start;writing-mode:lr-tb;text-anchor:start;fill:#0000ff" /></flowRegion><flowPara
|
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id="flowPara11317">$\beta$</flowPara><flowPara
|
||||
id="flowPara11319"></flowPara></flowRoot> </g>
|
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id="flowPara11317"
|
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style="font-size:40.00024796px;line-height:1.25">$\beta$</flowPara><flowPara
|
||||
id="flowPara11319"
|
||||
style="font-size:40.00024796px;line-height:1.25"> </flowPara></flowRoot> </g>
|
||||
</svg>
|
||||
|
|
|
|||
|
Before Width: | Height: | Size: 76 KiB After Width: | Height: | Size: 71 KiB |
|
|
@ -48,13 +48,16 @@
|
|||
{\bfseries}{}%
|
||||
{\newline}{}%
|
||||
\theoremstyle{break}
|
||||
|
||||
\newtheorem{lemma}{Lemma}[section]
|
||||
\newtheorem{fact}{Fact}[section]
|
||||
\newtheorem{corollary}{Corollary}[section]
|
||||
\newtheorem{proposition}{Proposition}[section]
|
||||
\newtheorem{example}{Example}[section]
|
||||
\newtheorem{problem}{Problem}[section]
|
||||
\newtheorem{definition}{Definition}[section]
|
||||
\newtheorem{theorem}{Theorem}[section]
|
||||
|
||||
\newcommand{\contradiction}{%
|
||||
\ensuremath{{\Rightarrow\mspace{-2mu}\Leftarrow}}}
|
||||
\newcommand*\quot[2]{{^{\textstyle #1}\big/_{\textstyle #2}}}
|
||||
|
|
@ -66,6 +69,10 @@
|
|||
|
||||
\newcommand{\sdots}{\smash{\vdots}}
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
\AtBeginDocument{\renewcommand{\setminus}{%
|
||||
\mathbin{\backslash}}}
|
||||
|
||||
|
|
@ -73,6 +80,8 @@
|
|||
\DeclareMathOperator{\Hom}{Hom}
|
||||
\DeclareMathOperator{\rank}{rank}
|
||||
\DeclareMathOperator{\Gl}{Gl}
|
||||
\DeclareMathOperator{\Lk}{lk}
|
||||
|
||||
|
||||
\titleformat{\section}{\normalfont \fontsize{12}{15} \bfseries}{%
|
||||
Lecture\ \thesection}%
|
||||
|
|
@ -263,12 +272,12 @@ Remark: there are knots that admit non isotopic Seifert surfaces of minimal genu
|
|||
|
||||
\begin{definition}
|
||||
Suppose $\alpha$ and $\beta$ are two simple closed curves in $\mathbb{R}^3$.
|
||||
On a diagram $L$ consider all crossings between $\alpha$ and $\beta$. Let $N_+$ be the number of positive crossings, $N_-$ - negative. Then the linking number: $lk(\alpha, \beta) = \frac{1}{2}(N_+ - N_-)$.
|
||||
On a diagram $L$ consider all crossings between $\alpha$ and $\beta$. Let $N_+$ be the number of positive crossings, $N_-$ - negative. Then the linking number: $\Lk(\alpha, \beta) = \frac{1}{2}(N_+ - N_-)$.
|
||||
\end{definition}
|
||||
\hfill
|
||||
\\
|
||||
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 $\beta$:
|
||||
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 $\beta$:
|
||||
\[
|
||||
\alpha \in H_1(S^3 \setminus \nu(\beta), \mathbb{Z}) \cong \mathbb{Z}.\]
|
||||
|
||||
|
|
@ -306,7 +315,7 @@ where $b_1$ is first Betti number of $\Sigma$.
|
|||
\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.
|
||||
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
|
||||
|
|
@ -581,19 +590,37 @@ g_{\big| \partial D^2} = f_{\big| \partial D^2.}
|
|||
&F: \mathbb{C}^2 \rightarrow \mathbb{C} \text{ a polynomial} \\
|
||||
&F(0) = 0
|
||||
\end{align*}
|
||||
Fact (Milnor Singular Points of Complex Hypersurfaces):
|
||||
\end{example}
|
||||
%\end{comment}
|
||||
|
||||
An oriented knot is called negative amphichiral if the mirror image $m(K)$ if $K$ is equivalent the reverse knot of $K$. \\
|
||||
\begin{example}[Problem]
|
||||
Prove that if $K$ is negative amphichiral, then $K \# K$ in
|
||||
$\mathbf{C}$
|
||||
\end{example}
|
||||
????????????
|
||||
\\
|
||||
\noindent
|
||||
as a corollary we see that $K_T^{n, }$ ???? \\
|
||||
is not slice unless $m=0$.
|
||||
\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}
|
||||
%
|
||||
%
|
||||
%
|
||||
\section{\hfill\DTMdate{2019-03-18}}
|
||||
\section{Concordance group \hfill\DTMdate{2019-03-18}}
|
||||
\begin{definition}
|
||||
A knot $K$ is called (smoothly) slice if $K$ is smoothly concordant to an unknot. \\
|
||||
A knot $K$ is smoothly slice if and only if $K$ bounds a smoothly embedded disk in $B^4$.
|
||||
|
|
@ -628,7 +655,7 @@ $K_1 \# K_2 \sim {K_1}^{\prime} \# {K_2}^{\prime}$.
|
|||
\fontsize{10}{10}\selectfont
|
||||
\centering{
|
||||
\def\svgwidth{\linewidth}
|
||||
\resizebox{0.8\textwidth}{!}{\input{images/concordance_sum.pdf_tex}}
|
||||
\resizebox{1\textwidth}{!}{\input{images/concordance_sum.pdf_tex}}
|
||||
}
|
||||
\caption{Sketch for Fakt \ref{fakt:concordance_connected}.}
|
||||
\label{fig:concordance_sum}
|
||||
|
|
@ -639,23 +666,40 @@ $K_1 \# K_2 \sim {K_1}^{\prime} \# {K_2}^{\prime}$.
|
|||
$K \# m(K) \sim $ the unknot.
|
||||
\end{fact}
|
||||
\noindent
|
||||
Let $\mathscr{C}$ denote all equivalent classes for knots. $\mathscr{C}$ is a group under taking connected sums, with neutral element (the class defined by) an unknot and inverse element (a class defined by) a mirror image.\\
|
||||
The figure eight knot is a torsion element in $\mathscr{C}$ ($2K \sim $ the unknot).\\
|
||||
\begin{example}[Problem]
|
||||
Are there in concordance group torsion elements that are not $2$ torsion elements? (open)
|
||||
\end{example}
|
||||
\begin{theorem}
|
||||
Let $\mathscr{C}$ denote a set of all equivalent classes for knots and $\{0\}$ denote class of all knots concordant to a trivial knot.
|
||||
$\mathscr{C}$ is a group under taking connected sums. The neutral element in the group is $\{0\}$ and the inverse element of an element $\{K\} \in \mathscr{C}$ is $-\{K\} = \{mK\}$.
|
||||
\end{theorem}
|
||||
\begin{fact}
|
||||
The figure eight knot is a torsion element in $\mathscr{C}$ ($2K \sim $ the unknot).
|
||||
\end{fact}
|
||||
\begin{problem}[open]
|
||||
Are there in concordance group torsion elements that are not $2$ torsion elements?
|
||||
\end{problem}
|
||||
\noindent
|
||||
Remark: $K \sim K^{\prime} \Leftrightarrow K \# -K^{\prime}$ is slice.
|
||||
|
||||
\\
|
||||
\\
|
||||
\noindent
|
||||
Let $\Omega$ be an oriented \\
|
||||
???????\\
|
||||
Suppose $\Sigma$ is a Seifert matrix with an intersection form ${(\alpha, \beta) \mapsto \Lk(\alpha, \beta^+)}$. Suppose $\alpha, \beta \in H_1(\Sigma, \mathbb{Z}$ (i.e. there are cycles). \\
|
||||
??????????????\\
|
||||
$\alpha, \beta \in \ker (H_1(\Sigma, \mathbb{Z}) \longrightarrow H_1(\Omega, \mathbb{Z}))$. Then there are two cycles $A, B \in \Omega$ such that $\partial A = \alpha$ and $\partial B = \beta$.
|
||||
Let $B^+$ be a push off of $B$ in the positive normal direction such that
|
||||
$\partial B^+ = \beta^+$.
|
||||
Then
|
||||
$\Lk(\alpha, \beta^+) = A \cdot B^+$
|
||||
%
|
||||
%
|
||||
\\
|
||||
\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{Poincaré duality}} H^2(X, \mathbb{Z} ) \xrightarrow{\text{evaluation}}\Hom(H_2(X, \mathbb{Z}), \mathbb{Z})
|
||||
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
|
||||
|
|
@ -749,7 +793,7 @@ In general
|
|||
|
||||
\section{\hfill\DTMdate{2019-05-20}}
|
||||
|
||||
Let $M$ be compact, oriented, connected four-dimensional manifold. If $H_1(M, \mathbb{Z}) = 0$ then there exists a
|
||||
Let $M$ be compact, oriented, connected four-dimensional manifold. If ${H_1(M, \mathbb{Z}) = 0}$ then there exists a
|
||||
bilinear form - the intersection form on $M$:
|
||||
|
||||
\begin{center}
|
||||
|
|
@ -1006,22 +1050,95 @@ 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.}}
|
||||
\noindent
|
||||
\begin{proof}
|
||||
By Poincar\'e duality we know that:
|
||||
\begin{align*}
|
||||
H_3(\Omega, Y) &\cong H^0(\Omega),\\
|
||||
H_2(Y) &\cong H^0(Y),\\
|
||||
H_2(\Omega) &\cong H^1(\Omega, Y),\\
|
||||
H_2(\Omega, Y) &\cong H^1(\Omega).
|
||||
\end{align*}
|
||||
Therefore $\dim_{\mathbb{Q}} \quot{H_1(Y)}{V}
|
||||
= \dim_{\mathbb{Q}} V
|
||||
$.
|
||||
\end{proof}
|
||||
\noindent
|
||||
Suppose $g(K) = 0$ ($K$ is slice). Then $H_1(\Sigma, \mathbb{Z}) \cong H_1(Y, \mathbb{Z})$. Let $g_{\Sigma}$ be the genus of $\Sigma$, $\dim H_1(Y, \mathbb{Z}) = 2g_{\Sigma}$. Then the Seifert form $V$ on 4\\
|
||||
?????\\
|
||||
has a subspace of dimension $g_{\Sigma}$ on which it is zero:
|
||||
\[
|
||||
\begin{pmatrix}
|
||||
0 & 0 & 0 \\
|
||||
\sdots & \sdots\makebox[0pt][l]{$\lbrce\left\lceil\frac i2\right\rceil$} & \sdots \\
|
||||
0 & 0 & \\
|
||||
\hline
|
||||
|
||||
& & 0 \\
|
||||
& & \makebox[0pt][r]{$\left\lfloor\frac i2\right\rfloor\rbrce$}\sdots \\
|
||||
0 & & 0
|
||||
\end{pmatrix}
|
||||
V =
|
||||
\begin{pmatrix}
|
||||
\begin{array}{c|c}
|
||||
0 & * \\
|
||||
\hline
|
||||
* & *
|
||||
\end{array}
|
||||
\end{pmatrix}
|
||||
\]
|
||||
\begin{align*}
|
||||
\newcommand*{\AddLeft}[1]{%
|
||||
\vadjust{%
|
||||
\vbox to 0pt{%
|
||||
\vss
|
||||
\llap{$%
|
||||
{#1}\left\{
|
||||
\vphantom{
|
||||
\begin{matrix}1\\\vdots\\0\end{matrix}
|
||||
}
|
||||
\right.\kern-\nulldelimiterspace
|
||||
\kern0.5em
|
||||
$}%
|
||||
\kern0pt
|
||||
}%
|
||||
}%
|
||||
}
|
||||
V = \qquad
|
||||
\begin{pmatrix}
|
||||
0 & \cdots & 0 & * & \cdots & * \\
|
||||
\vdots & & \vdots & \vdots & &\vdots \\
|
||||
0 & \cdots & 0 & * & \cdots & *
|
||||
\AddLeft{g_{\Sigma}}\\
|
||||
* & \cdots & * & * & \cdots & * \\
|
||||
\vdots & & \vdots & \vdots & &\vdots \\
|
||||
* & \cdots & * & * & \cdots & *
|
||||
\end{pmatrix}_{2g_{\Sigma} \times 2g_{\Sigma}}
|
||||
\end{align*}
|
||||
|
||||
\begin{align*}
|
||||
\newcommand\coolover[2]{\mathrlap{\smash{\overbrace{\phantom{%
|
||||
\begin{matrix} #2 \end{matrix}}}^{\mbox{$#1$}}}}#2}
|
||||
\newcommand\coolunder[2]{\mathrlap{\smash{\underbrace{\phantom{%
|
||||
\begin{matrix} #2 \end{matrix}}}_{\mbox{$#1$}}}}#2}
|
||||
\newcommand\coolleftbrace[2]{%
|
||||
#1\left\{\vphantom{\begin{matrix} #2 \end{matrix}}\right.}
|
||||
\newcommand\coolrightbrace[2]{%
|
||||
\left.\vphantom{\begin{matrix} #1 \end{matrix}}\right\}#2}
|
||||
\vphantom{% phantom stuff for correct box dimensions
|
||||
\begin{matrix}
|
||||
\overbrace{XYZ}^{\mbox{$R$}}\\ \\ \\ \\ \\ \\
|
||||
\underbrace{pqr}_{\mbox{$S$}}
|
||||
\end{matrix}}%
|
||||
\begin{matrix}% matrix for left braces
|
||||
\vphantom{a}\\
|
||||
\coolleftbrace{A}{e \\ y\\ y}\\
|
||||
\coolleftbrace{B}{y \\i \\ m}
|
||||
\end{matrix}%
|
||||
\begin{bmatrix}
|
||||
a & \coolover{R}{b & c & d} & x & \coolover{Z}{x & x}\\
|
||||
e & f & g & h & x & x & x \\
|
||||
y & y & y & y & y & y & y \\
|
||||
y & y & y & y & y & y & y \\
|
||||
y & y & y & y & y & y & y \\
|
||||
i & j & k & l & x & x & x \\
|
||||
m & \coolunder{S}{n & o} & \coolunder{W}{p & x & x} & x
|
||||
\end{bmatrix}%
|
||||
\begin{matrix}% matrix for right braces
|
||||
\coolrightbrace{x \\ x \\ y\\ y}{T}\\
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\coolrightbrace{y \\ y \\ x }{U}
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\end{matrix}
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||||
\end{align*}
|
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||||
|
||||
\end{comment}
|
||||
\end{document}
|
||||
|
|
|
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Reference in New Issue