The Polynomials of Prime Virtual Knots of Genus 1 and Complexity at Most 5

Abstract

Akimova and Matveev classified the prime virtual knots of genus 1 which admit diagrams with at most 5 classical crossings in 2017. In 2018, Kaur, Prabhakar, and Vesnin introduced the families of the $ L $ - and $ F $ -polynomials of virtual knots generalizing the Kauffman affine index polynomial. We introduce the notion of a totally flat-trivial virtual knot. We prove that the $ L $ - and $ F $ -polynomials for these knots coincide with the affine index polynomial. Also, we establish that all Akimova–Matveev knots are totally flat-trivial and calculate their affine index polynomials.