From 00d54747314a725a7ec7f9b4ebbd23133f2b380d Mon Sep 17 00:00:00 2001 From: Maria Marchwicka Date: Fri, 21 Jun 2019 23:24:32 -0500 Subject: [PATCH] small progress --- images/ball_4_pushed_seifert.pdf | Bin 4461 -> 4407 bytes images/ball_4_pushed_seifert.pdf_tex | 4 +- images/ball_4_pushed_seifert.svg | 63 +- images/figure8.svg | 39 +- images/satellite.pdf | Bin 12800 -> 12464 bytes images/satellite.pdf_tex | 4 +- images/satellite.svg | 40 +- images/satellite_v0.svg | 1209 ++++++++++++++++++++++++++ images/torus_alpha_beta.pdf | Bin 0 -> 6476 bytes images/torus_alpha_beta.pdf_tex | 59 ++ images/torus_alpha_beta.svg | 309 ++++++- lec_2.tex | 8 +- lec_3.tex | 16 +- lec_5.tex | 72 +- lectures_on_knot_theory.tex | 72 +- 15 files changed, 1785 insertions(+), 110 deletions(-) create mode 100644 images/satellite_v0.svg create mode 100644 images/torus_alpha_beta.pdf create mode 100644 images/torus_alpha_beta.pdf_tex diff --git a/images/ball_4_pushed_seifert.pdf 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fit-margin-left="0" + fit-margin-right="0" + fit-margin-bottom="0" /> @@ -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.}