The perturbed Alexander invariant ρ₁, 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 ρ₁ for families of knots {K_t} given by performing t full twists on a set of coherently oriented strands in a knot K₀ ⊂ S³. We prove that as t -> ∞ the coefficients of ρ₁ 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 ρ₁ to distinguish knots in infinite families, and we conjecture that ρ₁ 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.

This material is based upon work supported by the National Science Foundation under Award No. 2202704.