Polynomial invariants, knot homologies, and higher twist numbers of weaving knots W(3,n)

Abstract

We investigate several conjectures in geometric topology by assembling computer data obtained by studying weaving knots, a doubly infinite family [Formula: see text] of examples of hyperbolic knots. In particular, we compute some important polynomial knot invariants, as well as knot homologies, for the subclass [Formula: see text] of this family. We use these knot invariants to conclude that all knots [Formula: see text] are fibered knots and provide estimates for some geometric invariants of these knots. Finally, we study the asymptotics of the ranks of their Khovanov homology groups. Our investigations provide evidence for our conjecture that asymptotically as [Formula: see text] grows large, the ranks of Khovanov homology groups of [Formula: see text] are normally distributed.