The quandary of quandles: The Borel completeness of a knot invariant

TL;DR

This paper proves that determining isomorphism between knot quandles is a highly complex problem in descriptive set theory, highlighting its computational difficulty despite the knot class being manageable.

Contribution

It establishes that the quandle isomorphism problem is Borel complete, revealing its inherent complexity and contrasting it with the simpler classification of tame knots.

Findings

Isomorphism of quandles is Borel complete.

Tame knots form a trivial class in Borel reducibility.

The complexity of quandle isomorphism contrasts with the simplicity of classifying tame knots.

Abstract

The isomorphism type of the knot quandle introduced by Joyce is a complete invariant of tame knots. Whether two quandles are isomorphic is in practice difficult to determine; we show that this question is provably hard: isomorphism of quandles is Borel complete. The class of tame knots, however, is trivial from the perspective of Borel reducibility, suggesting that equivalence of tame knots may be reducible to a more tractable isomorphism problem.