# Polynomial Invariants, Knot Homologies, and Higher Twist Numbers of Weaving Knots $W(3,n)$

## TL;DR

This paper studies weaving knots W(3,n) by computing polynomial invariants and homologies, demonstrating they are fibered, estimating geometric invariants, and analyzing the asymptotic distribution of Khovanov homology ranks, providing evidence for a normal distribution conjecture.

## Contribution

It introduces new computational data on weaving knots W(3,n), proving they are fibered and exploring the asymptotic behavior of their Khovanov homology ranks.

## Key findings

- All W(3,n) knots are fibered.
- Estimated geometric invariants for W(3,n).
- Ranks of Khovanov homology tend to a normal distribution asymptotically.

## Abstract

We investigate several conjectures in geometric topology by assembling computer data obtained by studying weaving knots, a doubly infinite family $W(p,n)$ of examples of hyperbolic knots. In particular, we compute some important polynomial knot invariants, as well as knot homologies, for the subclass $W(3,n)$ of this family. We use these knot invariants to conclude that all knots $W(3,n)$ 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 $n$ grows large, the ranks of Khovanov homology groups of $W(3,n)$ are normally distributed.