# 2021-11-01

Tags: #web/quick-notes

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## 15:03 UGA Topology Seminar

Lev Tovstopyat-Nelip, “Floer Homology and Quasipositive Surfaces”, MSU.

• contact structure on an oriented 3-manifold $$Y$$: a maximally nonintegrable 2-place field $$\xi$$ where $$\xi = \ker( \alpha)$$ for some $$\alpha\in \Omega^1(Y)$$ with $$\alpha \vee d\alpha > 0$$.

• Example: $$\xi = \ker(\,dz+ r^2 \,d\theta)$$ on $${\mathbf{R}}^3$$.
• A transverse knot is a knot positively transverse to $${ \left.{{ \alpha }} \right|_{{K}} } > 0$$.

• Knots braided about the $$z{\hbox{-}}$$axis are naturally transverse, so study transverse knots via braids.
• A knot $$K \subseteq (Y, \xi)$$ is Legendrian if $${\mathbf{T}}K\leq { \left.{{\xi}} \right|_{{K}} }$$, so $${ \left.{{\alpha}} \right|_{{K}} } = 0$$.

• A disk $${\mathbb{D}}^2 \subseteq (Y, \xi)$$ is overtwisted if $${{\partial}}{\mathbb{D}}^2$$ is Legendrian, i.e. $${ \left.{{{\mathbf{T}}{\mathbb{D}}^2}} \right|_{{{{\partial}}{\mathbb{D}}^2}} } = { \left.{{\xi }} \right|_{{{\mathbb{D}}^2}} }$$.

• Eliashberg: overtwisted contact structures can be studied using algebraic topology (every homotopy class of plane fields contains an overtwisted contact structures)

• Tight contact structures are the interesting ones!

• Define self-linking number $$\sl(\widehat{B}) = w(B) - n$$.

• A type of adjunction inequality - If $$K$$ is transverse in $$(S^3, \xi_{\text{std}})$$ then $$\sl(K) \leq 2g(K) - 1$$.

• For $$\Sigma$$ an oriented surface with connected $$\phi\in {\operatorname{MCG}}(\Sigma, {{\partial}}\Sigma)$$, define $$Y_{\phi} \coloneqq S\times I / (x,1) \sim (\phi(x), 0)$$.

• Yields $$(\Sigma, \phi)$$ an open book decomposition.

• There is a correspondence between open book decompositions on $$Y$$ and contact structures on $$Y$$.

• Let $$\Sigma \hookrightarrow Y$$ be a Seifert surface, then it is quasipositive with respect to $$\xi$$ if there exists an o.b.d. $$(S, \phi)$$ such that $$\Sigma \subseteq S$$ is $$\pi_1{\hbox{-}}$$injective.

• Lyon: every Seifert surface in a closed oriented 3-manifold is quasipositive with respect to some contact structure $$\xi$$.

• Measure how far a knot is from being fibred: fibre depth.

• If $$K$$ is semi-quasipositive with respect to $$\xi_\text{std}$$, then $$\mkern 1.5mu\overline{\mkern-1.5mu\sl\mkern-1.5mu}\mkern 1.5mu(K) = 2g(K) - 1$$ Interesting #open/problems: does the converge hold? I.e. if $$\mkern 1.5mu\overline{\mkern-1.5mu\sl\mkern-1.5mu}\mkern 1.5mu(K) = 2g(K) - 1$$, is $$K$$ semi-quasipositive?

• $$\widehat{\operatorname{HFK}}$$ detects genus in the sense that $$g(K)$$ is the maximal nonvanishing $$\widehat{\operatorname{HFK}}(-S^3, K, i)$$.

• See fiberedness detection and sutured knot Floer homology.