Georgia Institute of Technology College of Sciences

This is a brief (15 minute) presentation of an undergraduate project that took place in the 2017 Fall semester.

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.

In this series of talks I will introduce branched coverings of manifolds and sketch proofs of most the known results in low dimensions (such as every 3 manifold is a 3-fold branched cover over a knot in the 3-sphere and the existence of universal knots). This week we continue discussing branched covers of 3-manifolds and prove universal links exist.

Heegaard Floer theory provides a powerful suite of tools for studying 3-manifolds and their subspaces. In 2006, Ozsvath, Szabo and Thurston defined an invariant of transverse knots which takes values in a combinatorial version of this theory for knots in the 3—sphere. In this talk, we discuss a refinement of their combinatorial invariant via branched covers and discuss some of its properties. This is joint work with Mike Wong.

The talk will include a crash course on infinite dimensional topology, with applications to various topological properties of the space of congruence classes of convex bodies in the Euclidean space.

In this series of talks I will introduce branched coverings of manifolds and sketch proofs of most the known results in low dimensions (such as every 3 manifold is a 3-fold branched cover over a knot in the 3-sphere and the existence of universal knots). This week we sstart discussing branched covers of 3-manifolds.

In this series of talks I will introduce branched coverings of manifolds and sketch proofs of most the known results in low dimensions (such as every 3 manifold is a 3-fold branched cover over a knot in the 3-sphere and the existence of universal knots). This week we should be able to finish our discussion of branched covers of surfaces and transition to 3-manifolds.

In this series of talks I will introduce branched coverings of manifolds and sketch proofs of most the known results in low dimensions (such as every 3 manifold is a 3-fold branched cover over a knot in the 3-sphere and the existence of universal knots). This week we will continue studying branched covers of surfaces. Among other things we should be able to see how to use branched covers to see some relations in the mapping class group of surfaces.

The immersed Seifert genus of a knot $K$ in $S^3$ can be defined as the minimal genus of an orientable immersed surface $F$ with $\partial F = K$. By a result of Gabai, this value is always equal to the (embedded) Seifert genus of $K$. In this talk I will discuss the embedded and immersed cross-cap numbers of a knot, which are the non-orientable versions of these invariants. Unlike their orientable counterparts these values do not always coincide, and can in fact differ by an arbitrarily large amount.