The Dabkowski-Sahi invariant and $4$-moves for links

TL;DR

This paper investigates the Dabkowski-Sahi invariant, a link invariant preserved under 4-moves, providing a new necessary condition to distinguish non-trivial links from trivial ones using this invariant.

Contribution

It introduces a practical obstruction criterion for determining when a link cannot be reduced to triviality via 4-moves based on the Dabkowski-Sahi invariant.

Findings

Provides a necessary condition for isomorphism of Dabkowski-Sahi invariants

Offers a practical obstruction to triviality under 4-moves

Enhances tools for link classification in knot theory

Abstract

Dabkowski and Sahi defined an invariant of a link in the $3$-sphere, which is preserved under $4$-moves. This invariant is a quotient of the fundamental group of the complement of the link. It is generally difficult to distinguish the Dabkowski-Sahi invariants of given links. In this paper, we give a necessary condition for the existence of an isomorphism between the Dabkowski-Sahi invariant of a link and that of the corresponding trivial link. Using this condition, we provide a practical obstruction to a link to be trivial up to $4$-moves.