Perturbed Gaussian generating functions for universal knot invariants

2021/09/05 by Bar-Natan, Dror, van der Veen, Roland · 4 citations
#57M25 #FOS: Mathematics #Geometric Topology (math.GT) #Quantum Algebra (math.QA)

paper · doi:10.48550/arxiv.2109.02057

Abstract

We introduce a new approach to universal quantum knot invariants that emphasizes generating functions instead of generators and relations. All the relevant generating functions are shown to be perturbed Gaussians of the form Pe^G, where G is quadratic and P is a suitably restricted "perturbation". After developing a calculus for such Gaussians in general we focus on the rank one invariant Z_\mathbbD in detail. We discuss how it dominates the \mathfraksl₂-colored Jones polynomials and relates to knot genus and Whitehead doubling. In addition to being a strong knot invariant that behaves well under natural operations on tangles Z_\mathbbD is also computable in polynomial time in the crossing number of the knot. We provide a full implementation of the invariant and provide a table in an appendix.

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