The 2-generalized knot group determines the knot

TL;DR

This paper proves that the 2-generalized knot group uniquely determines the unoriented knot type, extending the result to higher n-values, thereby linking algebraic invariants to knot classification.

Contribution

It establishes that the 2-generalized knot group fully determines the unoriented knot type and provides a sketch for the case of n>2, advancing knot invariant theory.

Findings

$G_2(K)$ determines the unoriented knot type

Sketch of proof for $G_n(K)$ with n>2

Strengthens the connection between generalized knot groups and knot classification

Abstract

Generalized knot groups $G_n(K)$ were introduced independently by Kelly (1991) and Wada (1992). We prove that $G_2(K)$ determines the unoriented knot type and sketch a proof of the same for $G_n(K)$ for $n>2$.