# Goussarov-Polyak-Viro's $n$-equivalence and the pure virtual braid group

## TL;DR

This paper extends Stanford's equivalence to virtual knots and string links using the lower central series of the pure virtual braid group, establishing its relation to Goussarov-Polyak-Viro's $n$-equivalence and finite type invariants.

## Contribution

It introduces the $L_n$-equivalence for virtual knots and links and proves its equivalence to the $n$-equivalence, linking it to finite type invariants.

## Key findings

- $L_n$-equivalence equals $n$-equivalence on virtual string links
- Two virtual string links are indistinguishable by finite type invariants of degree $n-1$ if $L_n$-equivalent
- Extension of Stanford's equivalence to virtual knot theory

## Abstract

In the context of finite type invariants, Stanford introduced a family of equivalence relations on knots defined by the lower central series of the pure braid groups and characterized the finite type invariants in terms of the structure of the braid groups. It is known that this equivalence and Ohyama's equivalence defined by a local move are equivalent. On the other hand, in the virtual knot theory, the concept of Ohyama's equivalence was extended by Goussarov-Polyak-Viro, which called an $n$-equivalence. In this paper we extend Stanford's equivalence to virtual knots and virtual string links by using the lower central series of the pure virtual braid group, and call it an $L_n$-equivalence. We then prove that the $L_n$-equivalence is equal to the $n$-equivalence on virtual string links. Moreover we directly prove that two virtual string links are not distinguished by any finite type invariants of degree $n-1$ if they are $L_n$-equivalent, for any positive integer $n$.