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 → ∞.