Twisted Knots and the Perturbed Alexander Invariant

The perturbed Alexander invariant, defined by Bar-Natan and van der Veen, is an infinite family of polynomial invariants of knots in the three-sphere. The first polynomial, rho_1, is quick to calculate and may be better at distinguishing knots than practically any other computable invariant; it also has deep connections to both classical and quantum topology. We will discuss the perturbed Alexander invariant and rho_1 in particular, and give results on the behavior of rho_1 and the classical Alexander polynomial under the operation of applying full twists to a knot. Our arguments use a model of random walks on a knot diagram.