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+ - + - + gradientTransform="matrix(0.15311825,0,0,0.14037805,-34.167106,63.536219)" /> + id="linearGradient2713" + x1="289.4693" + y1="1120.1073" + x2="282.93448" + y2="1180.0529" + gradientUnits="userSpaceOnUse" + gradientTransform="matrix(0.44914761,0,0,0.63606431,524.82916,14.169866)" /> + + + + fit-margin-bottom="18.1" /> + id="metadata5"> @@ -983,12 +1003,12 @@ inkscape:label="Layer 1" inkscape:groupmode="layer" id="layer1" - transform="translate(66.628853,-128.96645)"> + transform="translate(32.585366,-40.013863)"> + $\lambda$ ${\varphi_* (\lambda) =\lambda + \mu }$ $\mu$ $\mu$ + + + + sodipodi:nodetypes="sssss" /> + + + $\lambda$ $K$ $\mu$ + id="ellipse9417" + cx="0.16125841" + cy="65.959213" + rx="23.992804" + ry="10.68985" /> diff --git a/lec_5.tex b/lec_5.tex index db2b191..9cdf66b 100644 --- a/lec_5.tex +++ b/lec_5.tex @@ -114,9 +114,7 @@ So we can calculate: \end{proof} \begin{corollary} If $t$ is not a root of -$\det S S^T - $ \\ -????????????????\\ -then +$\det (tS - S^T) $, then $\vert \sigma_K(t) \vert \leq 2g$. \end{corollary} \begin{fact} @@ -157,49 +155,78 @@ was conjecture in the '70 and proved by P. Kronheimer and T. Mrówka (1994). \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{proposition} +Supremum of the signature function of the knot is bounded almost everywhere by two times $4$ - genus: +\[ +\ess \sup \vert \sigma_K(t) \vert \leq 2 g_4(K). +\] +\end{proposition} +\subsection{Topological genus} \begin{definition} -A knot $K$ is called topologically slice if $K$ bounds a topological locally flat disc in $B^4$ (it has tubular neighbourhood). +A knot $K$ is called topologically slice if $K$ bounds a topological locally flat disc in $B^4$ (i.e. the disk 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. +If $\Delta_K(t) = 1$, then $K$ is topologically slice (but not necessarily smoothly slice). \end{theorem} \begin{theorem}[Powell, 2015] -If $K$ is genus g -\\(top. loc.?????????)\\ +If $K$ is genus $g$ +(topologically flat) 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} -??????????????? +then \[ -H^1 (B^4 \setminus Y, \mathbb{Z}) = [B^4 \setminus Y, S^1] +\vert \sigma_K(t) - \sigma_{K^\prime}(t) \vert \leq 2 g \] +if $g_4^{\mytop}(K) \geq \ess \sup \vert \sigma_K(t) \vert$. +\end{theorem} +\noindent +The proof for smooth category was based on following equality: +\[ +\dim \ker (H_1 (Y) \longrightarrow H_1(\Omega)) = \frac{1}{2} \dim H_1(Y). +\] +For this equality we assumed that there exists a $3$ - dimensional manifold $\Omega$ (as shown in Figure \ref{fig:omega_in_B_4}) which was guaranteed by Pontryagin-Thom Construction.\\ +Pontryagin-Thom Construction relays on taking $\Omega$ as preimage of regular value: +\[ +H^1 (B^4 \setminus Y, \mathbb{Z}) = [B^4 \setminus Y, S^1], +\] +what relies on Sard's theorem, that the set of regular values has positive measure. But Sard's theorem doesn't work for topologically locally flat category. So there was a gap in the proof for topological locally flat category - the existence of $\Omega$.\\ \noindent Remark: unless $p=2$ or $p = 3 \wedge q = 4$: \[ -g_4^\top (T(p, q)) < q_4(T(p, q)) +g_4^{\mytop} (T(p, q)) < q_4(T(p, q)). \] -%?????????????????????? +% Wilczyński '93 +%Feller 2014 +%Baoder 2017 +%Lemark +\\ +\noindent +From the category of cobordant knots (or topologically cobordant knots) there exists a map to $\mathbb{Z}$ given by signature function. To any element $K$ we can associate a form +\[ +(1 - t)S + (1 - \bar{t})S^T) \in W(\mathbb{Z}[t, t^{-1}]). +\] This association is not well define because id depends on the choice of Seifert form. However, different choices lead ever to congruent forms ($S \mapsto CSC^T$) or induced the change on the form by adding or subtracting a hyperbolic element. \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 $(1 - t) S^\prime + (1 - t^{-1})S^T$. -%??????????????????????????? +$(1 - t) S + (1 - \bar{t}^{-1}) S^T$ 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: $A^T = A: (a, b) = \bar{(a, b)}$. \\ -???????????????????????????? +$ +W(\mathbb{Z}_p) = \mathbb{Z}_2 \oplus +\mathbb{Z}_2$ or +$\mathbb{Z}_4$ \\ +??????????????????????? +\\ +$\sum a_gt^j \longrightarrow \sum a_g t^{-1}$\\ \begin{theorem}[Levine '68] \[ -W(\mathbb{Z}[t^{\pm 1}) +W(\mathbb{Z}[t^{\pm 1}]) \longrightarrow \mathbb{Z}_2^\infty \oplus \mathbb{Z}_4^\infty \oplus \mathbb{Z} diff --git a/lectures_on_knot_theory.tex b/lectures_on_knot_theory.tex index 13561f7..57ff132 100644 --- a/lectures_on_knot_theory.tex +++ b/lectures_on_knot_theory.tex @@ -88,6 +88,7 @@ \DeclareMathOperator{\rank}{rank} \DeclareMathOperator{\coker}{coker} \DeclareMathOperator{\ord}{ord} +\DeclareMathOperator{\ess}{ess} \DeclareMathOperator{\mytop}{top} \DeclareMathOperator{\Gl}{GL} \DeclareMathOperator{\Sl}{SL} @@ -121,27 +122,28 @@ %\input{myNotes} \section{Basic definitions \hfill\DTMdate{2019-02-25}} -\input{lec_1.tex} +%\input{lec_1.tex} \section{Alexander polynomial \hfill\DTMdate{2019-03-04}} -\input{lec_2.tex} +%\input{lec_2.tex} %add Hurewicz theorem? \section{Examples of knot classes \hfill\DTMdate{2019-03-11}} -\input{lec_3.tex} +%\input{lec_3.tex} \section{Concordance group \hfill\DTMdate{2019-03-18}} -\input{lec_4.tex} +%\input{lec_4.tex} \section{Genus $g$ cobordism \hfill\DTMdate{2019-03-25}} -\input{lec_5.tex} +%\input{lec_5.tex} \section{\hfill\DTMdate{2019-04-08}} \input{lec_6.tex} \section{\hfill\DTMdate{2019-04-15}} +???????????????????\\ \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: @@ -236,6 +238,38 @@ H_1(Y, \mathbb{Z}) \times H_1(Y, \mathbb{Z}) &\longrightarrow \quot{\mathbb{Q}}{ A = V + V^T. \end{align*} ???????????????????????????? +\\ +We have a primary decomposition of $H_1(Y, \mathbb{Z}) = U$ (as a group). For any $p \in \mathbb{P}$ we define $U_p$ to be the subgroup of elements annihilated by the same power of $p$. We have $U = \bigoplus_p U_p$. +\begin{example} +\begin{align*} +\text{If } U &= +\mathbb{Z}_3 \oplus +\mathbb{Z}_{45} \oplus +\mathbb{Z}_{15} \oplus +\mathbb{Z}_{75} +\text{ then }\\ +U_3 &= +\mathbb{Z}_3 \oplus +\mathbb{Z}_9 \oplus +\mathbb{Z}_3 \oplus +\mathbb{Z}_3 +\text{ and }\\ +U_5 &= +(e) \oplus +\mathbb{Z}_5 \oplus +\mathbb{Z}_5 \oplus +\mathbb{Z}_{25}. +\end{align*} +\end{example} + +\begin{lemma} +Suppose $x \in U_{p_1}$, $y \in U_{p_2}$ and $p_1 \neq p_2$. Then $ = 0$. +\end{lemma} +\begin{proof} +\begin{align*} +x \in U_{p_1} +\end{align*} +\end{proof} \begin{align*} H_1(Y, \mathbb{Z}) \cong \quot{\mathbb{Z}^n}{A\mathbb{Z}}\\ A \longrightarrow BAC^T \quad \text{Smith normal form} @@ -511,7 +545,7 @@ $H_1(\bar{X}$ field of fractions -\section{\hfill\DTMdate{2019-06-03}} +\section{Surgery \hfill\DTMdate{2019-06-03}} \begin{theorem} Let $K$ be a knot and $u(K)$ its unknotting number. Let $g_4$ be a minimal four genus of a smooth surface $S$ in $B^4$ such that $\partial S = K$. Then: \[ @@ -521,9 +555,9 @@ u(K) \geq g_4(K) Recall that if $u(K)=u$ then $K$ bounds a disk $\Delta$ with $u$ ordinary double points. \\ \noindent -Remove from $\Delta$ the two self intersecting and glue the Seifert surface for the Hopf link. The reality surface $S$ has Euler characteristic $\chi(S) = 1 - 2u$. Therefore $g_4(S) = u$ . +Remove from $\Delta$ the two self intersecting disks and glue the Seifert surface for the Hopf link. The reality surface $S$ has Euler characteristic $\chi(S) = 1 - 2u$. Therefore $g_4(S) = u$. \end{proof} -???????????????????\\ +%Tim D. Cochran and Peter Teichner \begin{example} The knot $8_{20}$ is slice: $\sigma \equiv 0$ almost everywhere but $\sigma(e^{\frac{ 2\pi i}{6}}) = + 1$. \end{example} @@ -532,8 +566,8 @@ The knot $8_{20}$ is slice: $\sigma \equiv 0$ almost everywhere but $\sigma(e^{\ %Journal-ref: Comment. Math. Helv. 79 (2004) 105-123 \subsection*{Surgery} %Rolfsen, geometric group theory, Diffeomorpphism of a torus, Mapping class group -Recall that $H_1(S^1 \times S^1, \mathbb{Z}) = \mathbb{Z}^3$. As generators for $H_1$ we can set ${\alpha = [S^1 \times \pt]}$ and ${\beta=[\pt \times S^1]}$. Suppose ${\phi: S^1 \times S^1 \longrightarrow S^1 \times S^1}$ is a diffeomorphism. -Consider an induced map on homology group: +Recall that $H_1(S^1 \times S^1, \mathbb{Z}) = \mathbb{Z}^2$. As generators for $H_1$ we can set ${\alpha = [S^1 \times \pt]}$ and ${\beta=[\pt \times S^1]}$. Suppose ${\phi: S^1 \times S^1 \longrightarrow S^1 \times S^1}$ is a diffeomorphism. +Consider an induced map on the homology group: \begin{align*} H_1(S^1 \times S^1, \mathbb{Z}) \ni \phi_* (\alpha) &= p\alpha + q \beta, \quad p, q \in \mathbb{Z},\\ \phi_*(\beta) &= r \alpha + s \beta, \quad r, s \in \mathbb{Z}, \\ @@ -541,11 +575,10 @@ H_1(S^1 \times S^1, \mathbb{Z}) \ni \phi_* (\alpha) &= p\alpha + q \beta, \quad \begin{pmatrix} p & q\\ r & s - \end{pmatrix} + \end{pmatrix}. \end{align*} As $\phi_*$ is diffeomorphis, it must be invertible over $\mathbb{Z}$. Then for a direction preserving diffeomorphism we have $\det \phi_* = 1$. Therefore $\phi_* \in \Sl(2, \mathbb{Z})$. \end{theorem} - \vspace{10cm} \begin{theorem} Every such a matrix can be realized as a torus. @@ -564,6 +597,15 @@ S^1 \times \pt &\longrightarrow \pt \times S^1 \\ \item \end{enumerate} \end{proof} +\begin{figure}[h] +\fontsize{20}{10}\selectfont +\centering{ +\def\svgwidth{\linewidth} +\resizebox{0.5\textwidth}{!}{\input{images/dehn_twist.pdf_tex}} +\caption{Dehn twist.} +\label{fig:dehn_twist} +} +\end{figure}