A surface construction for colored Khovanov homology

Christine Lee (South Alabama)

Abstract

Colored Khovanov homology is a categorification of the colored Jones polynomial. To each integer \(n ≥ 2\) and a diagram \(D\) of a link, it assigns a bigraded chain complex \(\{C^{Kh}_{i,j} (D, n)\}\). The graded Euler characteristic of the homology groups \(\{H^{Kh}_{i,j} (D, n)\}\) gives the nth colored Jones polynomial. It has typically been difficult to extract topological information from colored Khovanov homology due to its dependence on the combinatorics of link diagrams. We will give a construction of colored Khovanov homology of a knot in terms of embedded surfaces in the complement to more intrinsically motivate it using topology, and we will discuss potential applications. This work draws inspiration from Bar-Natan’s formulation of Khovanov homology and the Strong Slope Conjecture by Garoufalidis and Kalfagianni-Tran relating the colored Jones polynomial to topology of essential surfaces in the knot complement.