Ineffectiveness of homotopical invariants on Nakanishi's 4-move conjecture

TL;DR

This paper demonstrates that homotopical invariants derived from the fundamental group are ineffective in distinguishing knots under Nakanishi's 4-move conjecture, suggesting limitations of such invariants for unknotting problems.

Contribution

The paper proves that any knot invariant based on the fundamental group and preserved by 4-moves remains constant, indicating these invariants cannot resolve Nakanishi's 4-move conjecture.

Findings

Homotopical invariants are invariant under 4-moves.

Such invariants cannot distinguish knots related by 4-moves.

The Dabkowski-Sahi invariant is constant under 4-moves.

Abstract

A $4$-move is a local operation for links consisting in replacing two parallel arcs by four half twists. At the present time, it is not known if this induces an unkotting operation for knots. Studying the Dabkowski-Sahi invariant, we prove that any invariant of knots based on the fundamental group $\pi_1(S^3\setminus K)$ and preserved by $4$-moves is constant among the isotopy classes of knots.