The Jones Polynomial and Khovanov Homology of Weaving Knots $W(3,n)$

Abstract

In this paper we compute the signature for a family of knots $W(k,n)$, the weaving knots of type $(k,n)$. By work of E.~S.~Lee the signature calculation implies a vanishing theorem for the Khovanov homology of weaving knots. Specializing to knots $W(3,n)$, we develop recursion relations that enable us to compute the Jones polynomial of $W(3,n)$. Using additional results of Lee, we compute the ranks of the Khovanov Homology of these knots. At the end we provide evidence for our conjecture that, asymptotically, the ranks of Khovanov Homology of $W(3,n)$ are {\it normally distributed}.