Hecke algebra trace algorithm and some conjectures on weave knots

Abstract

In this paper, we perform a case study of a very special family of knots \(W(p,m)\) known as {\it weaving knots of type \((p,m)\).} This infinite family of knots are hyperbolic and alternating and there are known interesting results as well as conjectures which relate their topological invariants to their geometric invariants. To explore more on the relationship between the topological and geometric invariants of this family of knots, we look upon these knots as closure of \(p\) braids written in terms of generators of braid group \(B_p\) and consider the image of this braid under representation of \(B_p\) into the Hecke algebra \(H_p(q)\) over a field \(K\). We write an algorithm to explicitly compute the trace of this element in \(H_p(q)\) corresponding to parameters \(q\) and \(z.\) We use this algorithm to derive invariants of these knots. We generate data for \(W(4,m)\), \(W(5,m)\) and \(W(6,m)\) families and provide evidence for the conjecture of Dasbach and Lin that higher twist numbers defined by them in \cite{DasbachLin} provide better volume bound for weaving knots than in \cite{Weaving_vol}. We also look at the asymptotics of the distribution of ranks of Khovanov homology groups of \(W(p,m)\) for a fixed \(p\) as \(m\) grows large.