Morse matchings and Khovanov homology of 4-strand torus links

Abstract

Given a link or a tangle diagram, we define algorithmic Morse theoretic simplifications on their Khovanov homology. In contrast to Bar-Natan's scanning algorithm, the cancellations are postponed until the end and performed in one go. Although our novel approach is computationally inferior to Bar-Natan's algorithm, it side-steps the need for a large amount of iterations, making it more fitting for theoretical analysis. Our main application is towards integral Khovanov homology of 4-strand torus links, for which we compute non-trivial Khovanov homology groups in all homological degrees and find an abundance of $4$-torsion. At the limit $T(4,\infty )$, our computations agree with a conjecture of Gorsky, Oblomkov and Rasmussen. For finite $n$, we use the $\lambda$-invariant of Lewark, Marino and Zibrowius to derive lower bounds on proper rational Gordian distances from $T(4,n)$.

53 pages, many figures. Comments welcome!