Georgia Institute of Technology College of Sciences

In this talk I will present a complete set of Legendrian Kirby moves relating diagrams of the same contact 3–manifold, yielding a contact analogue of Kirby's theorem. Our approach refines Avdek's ribbon moves by using Gervais' presentation of mapping class groups using only nonseparating curves, together with explicit constructions of Legendrian representatives. This is a joint work with with Marc Kegel and Eric Stenhede.

A contact 3-manifold is planar if it is compatible with an open book whose page genus is 0. Wendl's theorem that any symplectic filling of a planar contact 3-manifold comes as a Lefschetz fibration filling a given planar open book translates questions about symplectic fillings of planar contact 3-manifolds into group theoretical questions about Dehn twist factorizations in the planar mapping class groups. I will outline work with Matt Day studying the nilpotent levels and truncations of the planar mapping class groups. We have calculated the ranks of the first three levels in the lower central series. I will explain these calculations and will outline a procedure to determine whether a mapping class element has a positive factorization. This work is joint with Matt Day.

It has been observed that the relationship of ribbon concordance induces monotonic behavior across several link invariants. 

Resolving a famous open question of Gordon, Agol unified these observations by showing that ribbon concordance induces a partial order on knots in the 3-sphere.

Agol's argument can be extended to show that ribbon concordance likewise induces a partial order on arbitrary links in the 3-sphere.

Perhaps motivated by the slice-ribbon conjecture, it is natural and interesting to ask which knots and links are minimal with respect to this partial order. 

Leveraging recent developments in link Floer homology, I will share new examples of ribbon minimal knots and links. 

Some of this work is joint with Jaewon Lee and Alessio Di Prisa.

Given a link in a closed 3-manifold, which Dehn surgery multislopes give rise to 3-manifolds with taut foliations? In this talk, I will discuss the ziggurat phenomenon: restricting to those foliations transverse to a fixed flow on the link complement, the set of multislopes typically has a fractal staircase shape with rational corners. In joint work with Thomas Massoni, we explain the ziggurat phenomenon in some contexts using tools from contact geometry. For those who have already seen a talk about this work, I'll try to emphasize some different aspects.