Thesis Defense - Mary Stelow

Speaker: Mary Stelow

Title:  Towards Combinatorial Khovanov Invariants for Admissible Links in $S^1 \times S^2$ using Instantons

 

Abstract (in TeX): Khovanov homology is a powerful invariant for links in $S^3$. There has been recent interest in building extensions of Khovanov homology for links in other 3-manifolds. Given an admissible link $L$ inside a 3-manifold $Y$, one proposal to do this is to use the spectral sequences converging to $I_*(Y, L)$ described by Kronheimer and Mrowka in their celebrated unknot detection proof.  The early pages of these spectral sequences suggest candidates for Khovanov-style invariants of links in $Y$, and in certain 3-manifold–link settings, these have been shown to recover existing extensions of Khovanov homology. This thesis addresses the question of whether a Khovanov invariant for admissible links in $Y = S^1 \times S^2$ could be developed using this procedure.

 

To this end, we develop instanton calculations and techniques to compute the $E_2$ pages of certain Kronheimer-Mrowka spectral sequences for admissible links $L \subset S^1 \times S^2$ converging to $I_*(S^1 \times S^2, L)$. In doing so, we discover a new annular Khovanov-style invariant $\text{Cube}(\cdot)$ for admissible links in $S^1 \times D^2$, and investigate its structural properties. We then explore the extent to which $\text{Cube} (\cdot)$ may be used to define combinatorial link invariants in $S^1 \times S^2$.  We produce examples to show that the naive approach to defining a link theory in this manner does not produce an invariant. To the author's knowledge, this is the first instance of a setting where the $E_2$ page is not a link invariant. We provide some speculations on structural reasons for this failure, and discuss how this work relates to existing proposals for Khovanov homology in $S^1\times S^2$. Throughout, we explore many new phenomena and also reinterpret some recent results in the literature, and we develop several new techniques for calculation.