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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.

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