Narrow Results

This note gives an explicit calculation of the doubly infinite sequence Δ(p, q, 2m), m ∈ Z of Alexander polynomials of the (p, q) torus knot with m extra full twists on two adjacent strings, where p and q are both positive. The knots can be presented as the closure of the p-string braids , where δ_p = σ_p-1σ_p-2 · σ₂σ₁, or equally of the q-string braids . As an application we give conditions on (p, q) which ensure that all the polynomials Δ(p, q, 2m) with |m| ≥ 2 have at least one coefficient a with |a| > 1. A theorem of Ozsvath and Szabo then ensures that no lens space can arise by Dehn surgery on any of these knots. The calculations depend on finding a formula for the multivariable Alexander polynomial of the 3-component link consisting of the torus knot with twists and the two core curves of the complementary solid tori.

The normal holonomy of a smooth knot in Euclidean space is a geometric invariant with value in the unit circle which is closely related to the writhing number. Here we prove that normal holonomy fibers the space of smooth knots, except for possible singularities on the set of round knots. We deduce results on the existence of isotopies of constant holonomy, writhe and twist.

Detailed analyses and proofs of various formulae used for the calculation and estimation of the writhe of a space curve are given. A new formula for calculating the rate of change of writhe under smooth deformation is presented. This latter result is used to show that the writhe of a closed curve evolving under certain nonlinear evolution equations is conserved.