Generalised knot groups distinguish the square and granny knots (with an appendix by David Savitt)

TL;DR

This paper demonstrates that generalized knot groups G_n(K) can distinguish between the square and granny knots, which have identical fundamental groups, thus providing a more refined knot invariant.

Contribution

It proves that for all n>1, the groups G_n(SK) and G_n(GK) are non-isomorphic, confirming a conjecture and showing these groups contain more information than the fundamental group.

Findings

G_n(SK) and G_n(GK) are non-isomorphic for all n>1

The isomorphism type of G_n(K) encodes more knot information than the fundamental group

Confirmed a conjecture of Lin and Nelson regarding generalized knot groups

Abstract

Given a knot K we may construct a group G_n(K) from the fundamental group of K by adjoining an nth root of the meridian that commutes with the corresponding longitude. These "generalised knot groups" were introduced independently by Wada and Kelly, and contain the fundamental group as a subgroup. The square knot SK and the granny knot GK are a well known example of a pair of distinct knots with isomorphic fundamental groups. We show that G_n(SK) and G_n(GK) are non-isomorphic for all n>1. This confirms a conjecture of Lin and Nelson, and shows that the isomorphism type of G_n(K), n>1, carries more information about K than the isomorphism type of the fundamental group. An appendix by David Savitt contains some results on representations of the trefoil group in PSL(2,p) that are needed for the proof.