2024/03/06 by Joe Boninger, Boninger, Joe
Materials Science · Mathematics · #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Mathematical Dynamics and Fractals #Quantum Algebra (math.QA) #Quasicrystal Structures and Properties
paper · pdf · doi:10.48550/arxiv.2403.03754
openalex publication_date 2024/03/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 \Kt\ 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.