# Goussarov-Polyak-Viro Conjecture for degree three case

## TL;DR

This paper explicitly constructs seven Gauss diagram formulas for degree three Vassiliev invariants of long virtual knots, confirming the Goussarov-Polyak-Viro conjecture for this case.

## Contribution

It provides explicit formulas for all degree three invariants and proves their correspondence with classical knot invariants, advancing understanding of virtual knot invariants.

## Key findings

- Seven Gauss diagram formulas for degree three invariants are explicitly given.
- The Polyak-Viro formula is included among the classical knot formulas.
- The Goussarov-Polyak-Viro conjecture is confirmed for degree three invariants.

## Abstract

Although it is known that the dimension of the Vassiliev invariants of degree three of long virtual knots is seven, the complete list of seven distinct Gauss diagram formulas have been unknown explicitly, where only one known formula was revised without proof. In this paper, we give seven Gauss diagram formulas to present the seven invariants of the degree three (Proposition 4). We further give 23 Gauss diagram formulas of classical knots (Proposition 5). In particular, the Polyak-Viro Gauss diagram formula [19] is not a long virtual knot invariant; however, it is included in the list of 23 formulas. It has been unknown whether this formula would be available by arrow diagram calculus automatically. In consequence, as it relates to the conjecture of Goussarov-Polyak-Viro [8, Conjecture 3.C], for all the degree three finite type long virtual knot invariants, each Gauss diagram formula is represented as those of Vassiliev invariants of classical knots (Theorem 1).