Twisted Knots and the Perturbed Alexander Invariant

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 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. Our methods also yield a new proof that the Alexander polynomial stabilizes under such twisting and we show how to compute the limit of Alexander polynomials, strengthening a recent result of Chen. Finally, we conjecture that ρ₁ obstructs knot positivity via a ``perturbed Conway invariant.''