Georgia Institute of Technology College of Sciences

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.

Engel structures are maximally non-integrable rank-two plane fields on four-dimensional manifolds. They are closely related to contact geometry, but their global behavior is still much less understood.

In contact topology, complex tangencies of real hypersurfaces in complex manifolds give a fundamental source of contact structures, often with strong rigidity properties. This motivates the Engel analogue: can a compact four-dimensional submanifold of $\mathbb C^3$ have complex tangencies forming an Engel structure?

In this talk, I will explain how to construct such examples in the case of embeddings $M \times S^1 \subset \mathbb C^3$. The main idea is to start from a standard construction of Engel structures on circle bundles over $3$-manifolds, and then realize these Engel distributions as complex tangencies of a suitable embedding into $\mathbb C^3$.  This gives the first compact examples of submanifolds of $\mathbb C^3$ whose complex tangencies are Engel, answering a question of Yakov Eliashberg. This is joint work with E. Fernández and Á. del Pino.