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

Rama Mishra, Ross Staffeldt

Published 2019-02-05Version 1

In this paper we first compute the signature for a family of knots $W(k,n)$, the weaving knots of type $(k,n)$. Specializing to knots $W(3,n)$ we develop recursive formulas for elements in the Hecke algebras arising from representations of $W(3,n)$ as the closure of braids on three strands. These formulas enable us to gather information about polynomial invariants such as the HOMFLY-PT, Jones, and Alexander polynomials of $W(3,n)$. Using the coefficients of Alexander polynomial along with the signature information and the fact that these knots are alternating, we compute the Heegard-Floer homology groups. We prove that the Seifert genus $g(W(3,n))=n{-}1$ and also conclude that each $W(3,n)$ is a fibered knot. From the Jones polynomial we compute the higher twist numbers for this family of knots and investigate their relation to the hyperbolic volume of the complement of the knot. We also use the Jones polynomial to compute rational, as well as integral, Khovanov homology groups of $W(3,n)$. 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.