Polynomials of genus one prime knots of complexity at most five

TL;DR

This paper classifies genus one prime knots with up to five crossings, introduces new polynomial invariants for virtual knots, and shows that certain knots have coinciding polynomial invariants.

Contribution

It proves all Akimova-Matveev genus one prime knots are totally flat-trivial and computes their affine index polynomials, linking classical and virtual knot invariants.

Findings

Akimova-Matveev knots are totally flat-trivial

F-polynomials and L-polynomials coincide with affine index polynomial for these knots

Explicit affine index polynomials are calculated for all such knots

Abstract

Prime knots of genus one admitting diagram with at most five classical crossings were classified by Akimova and Matveev in 2014. In 2018 Kaur, Prabhakar and Vesnin introduced families of L-polynomials and F-polynomials for virtual knots which are generalizations of affine index polynomial. Here we introduce a notion of totally flat-trivial knots and demonstrate that for such knots F-polynomials and L-polynomials coincide with affine index polynomial. We prove that all Akimova - Matveev knots are totally flat-trivial and calculate their affine index polynomials.