The Khovanov homology of (p,-p,q) pretzel knots

TL;DR

This paper computes the Khovanov homology over  for (p,-p,q) pretzel knots with odd p and large q, revealing they have thin homology and are non-quasi-alternating, expanding understanding of knot invariants.

Contribution

It provides explicit computations and a conjecture for the Khovanov homology of (p,-p,q) pretzel knots, identifying a new class of non-quasi-alternating knots with thin homology.

Findings

Knots have thin Khovanov homology over  and z.

Computed for p=3 to 15 and large q.

Identifies an infinite class of non-quasi-alternating knots with thin homology.

Abstract

In this paper, we compute the Khovanov homology over \Q for (p,-p,q) pretzel knots for odd values of p from 3 to 15 and arbitrarily large q. We provide a conjecture for the general form of the Khovanov homology of (p,-p,q) pretzel knots. These computations reveal that these knots have thin Khovanov homology (over \Q and \Z). Because Greene has shown that these knots are not quasi-alternating, this provides an infinite class of non-quasi-alternating knots with thin Khovanov homology.