Twisted Knots and the Perturbed Alexander Invariant

The perturbed Alexander invariant $\rho_1$, defined by Bar-Natan and van der Veen, is a powerful, easily computable polynomial knot invariant with deep connections to the Alexander and colored Jones polynomials. We study the behavior of $\rho_1$ for families of knots $\{K_t\}$ given by performing $t$ full twists on a set of coherently oriented strands in a knot $K_0 \subset S^3$. We prove that as $t \to \infty$ the coefficients of $\rho_1$ grow asymptotically linearly, and we show how to compute this growth rate for any such family. As an application we give the first theorem on the ability of $\rho_1$ to distinguish knots in infinite families, and we conjecture that $\rho_1$ obstructs knot positivity via a "perturbed Conway invariant." Along the way we expand on a model of random walks on knot diagrams defined by Lin, Tian and Wang.