Georgia Institute of Technology College of Sciences

It is generally a difficult problem to compute the Betti numbers of a given finite-index subgroup of an infinite group, even if the Betti numbers of the ambient group are known. In this talk, I will describe a procedure for obtaining new lower bounds on the first Betti numbers of certain finite-index subgroups of the braid group. The focus will be on the level 4 braid group, which is the kernel of the mod 4 reduction of the integral Burau representation. This is joint work with Dan Margalit.

Peter Lambert-Cole: Mutant knots are notoriously hard to distinguish. Many, but not all, knot invariants take the same value on mutant pairs. Khovanov homology with coefficients in Z/2Z is known to be mutation-invariant, while the bigraded knot Floer homology groups can distinguish mutants such as the famous Kinoshita-Terasaka and Conway pair. However, Baldwin and Levine conjectured that delta-graded knot Floer homology, a singly-graded reduction of the full invariant, is preserved by mutation. In this talk, I will give a new proof that Khovanov homology mod 2 is mutation-invariant. The same strategy can be applied to delta-graded knot Floer homology and proves the Baldwin-Levine conjecture for mutations on a large class of tangles. -----------------------------------------------------------------------------------------------------------------------------------------------Alex Zupan: Generally speaking, given a type of manifold decomposition, a natural problem is to determine the structure of all decompositions for a fixed manifold. In particular, it is interesting to understand the space of decompositions for the simplest objects. For example, Waldhausen's Theorem asserts that up to isotopy, the 3-sphere has a unique Heegaard splitting in every genus, and Otal proved an analogous result for classical bridge splittings of the unknot. In both cases, we say that these decompositions are "standard," since they can be viewed as generic modifications of a minimal splitting. In this talk, we examine a similar question in dimension four, proving that -- unlike the situation in dimension three -- the unknotted 2-sphere in the 4-sphere admits a non-standard bridge trisection. This is joint work with Jeffrey Meier.

In a joint work with M. Yampolsky, we gave a classification of Thurston maps with parabolic orbifolds based on our previous results on characterization of canonical Thurston obstructions. The obtained results yield a solution to the problem of algorithmically checking combinatorial equivalence of two Thurston maps.

Every four-dimensional Stein domain has a Morse function whoseregular level sets are contact three-manifolds. This allows us to studycomplex curves in the Stein domain via their intersection with thesecontact level sets, where we can comfortably apply three-dimensional tools.We use this perspective to understand links in Stein-fillable contactmanifolds that bound complex curves in their Stein fillings.