Abstract: Colored Khovanov homology is a categorification of the colored Jones polynomial. To each integer $n \geq 2$ 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. Inspired by Bar-Natan’s formulation of Khovanov homology for tangles and other approaches to topological formulations for Khovanov homology by McDougall and Seidel-Smith, 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.
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