Goussarov-Polyak-Viro combinatorial formulas for finite type invariants

TL;DR

This paper simplifies the proof that finite type invariants of knots can be expressed through local combinatorial formulas, and extends these results to tangles and braids.

Contribution

It provides a simplified approach to Goussarov-Polyak-Viro formulas and generalizes them to pure tangles and braids.

Findings

Existence of Gauss diagram formulas for finite type invariants.

Simplified proof approach for multi-local formulas.

Extension to pure tangles and braids.

Abstract

Goussarov, Polyak, and Viro proved that finite type invariants of knots are ``finitely multi-local'', meaning that on a knot diagram, sums of quantities, defined by local information, determine the value of the knot invariant. The result implies the existence of Gauss diagram combinatorial formulas for finite type invariants. This article presents a simplified account of the original approach. The simplifications provide an easy generalization to the cases of pure tangles and pure braids. The associated problem on group algebras is introduced and used to prove the existence of ``multi-local word formulas'' for finite type invariants of pure braids.