Constructions and Computations in Khovanov Homology

Abstract

In this thesis, we present a collection of results relating to Khovanov homology.

We consider the family of 3-strand pretzel links, and compute their unreduced and reduced

Khovanov homology using two different methods. We also show how to extend

Lawrence Roberts’ totally twisted Khovanov homology to integer coefficients, yielding

a spanning tree model for odd Khovanov homology with an explicitly computable

differential. Finally, we show that Khovanov’s functor-valued invariant of tangles

contains the same information as Bar-Natan’s dotted cobordism tangle theory, and

we construct a natural bordered theory for Khovanov homology using this invariant.