---
date: 2021-11-01
tags: [ web/quick-notes ]
---

# 2021-11-01

Tags:
#web/quick-notes

Refs:
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## 15:03 UGA Topology Seminar

> Lev Tovstopyat-Nelip, "Floer Homology and Quasipositive Surfaces", MSU.

- [contact structure](contact%20structure) on an oriented [3-manifold](3-manifold) $Y$: a maximally nonintegrable 2-place field $\xi$ where $\xi = \ker( \alpha)$ for some $\alpha\in \Omega^1(Y)$ with $\alpha \wedgeprod d\alpha > 0$.
  - Example: $\xi = \ker(\dz + r^2 \dtheta)$ on $\RR^3$.

- A **transverse knot** is a knot positively transverse to $\ro{ \alpha }{K} > 0$.
  - Knots braided about the $z\dash$axis are naturally transverse, so study transverse knots via braids.

- A knot $K \subseteq (Y, \xi)$ is [Legendrian](Legendrian) if $\T K\leq \ro{\xi}{K}$, so $\ro{\alpha}{K} = 0$.

- A disk $\DD^2 \subseteq (Y, \xi)$ is [overtwisted](overtwisted) if $\bd \DD^2$ is Legendrian, i.e. $\ro{\T \DD^2}{\bd \DD^2} = \ro \xi {\DD^2}$.


- 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 $\sl(\hat B) = w(B) - n$.
- A type of [adjunction inequality](adjunction%20inequality) - If $K$ is transverse in $(S^3, \xi_{\std})$ then $\sl(K) \leq 2g(K) - 1$.

- For $\Sigma$ an oriented surface with connected $\phi\in \MCG(\Sigma, \bd \Sigma)$, define $Y_{\phi} \da S\cross I / (x,1) \sim (\phi(x), 0)$.

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

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

- Let $\Sigma \embeds Y$ 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 $\pi_1\dash$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](fibred): **fibre depth**.

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

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

- See *fiberedness detection* and *sutured knot Floer homology*.