Jones Polynomials and their Zeros for a Family of Knots and Links

We calculate Jones polynomials V(Hr,t) for a family of alternating knots and links Hr with arbitrarily many crossings r, by computing the Tutte polynomials T(G+(Hr),x,y) for the associated graphs G+(Hr) and evaluating these with x=-t and y=-1/t. Our method enables us to circumvent the generic feature that the computational complexity of V(Lr,t) for a knot or link Lr for generic t grows exponentially rapidly with r. We also study the accumulation set of the zeros of these polynomials in the limit of infinitely many crossings, r→∞.