Polyak type equations for virtual arrow diagram formulas in the annulus

TL;DR

This paper characterizes the space of arrow diagram formulas for virtual knots in the annulus using linear algebra, leading to a refinement of existing theorems on planar chain invariants.

Contribution

It introduces a linear algebraic framework for arrow diagram formulas in the annulus, inspired by Polyak's conjecture, and improves a key theorem in knot invariants.

Findings

Characterization of arrow diagram formulas as a kernel of a linear map

Refinement of Grishanov-Vassiliev's theorem for planar chain invariants

Enhanced understanding of virtual knot invariants in the annulus

Abstract

We describe the space of arrow diagram formulas for virtual knot diagrams in the annulus as the kernel of a linear map, inspired from a conjecture due to M. Polyak. As a main application, we slightly improve Grishanov-Vassiliev's theorem for planar chain invariants.