Test sets for: Hard unknots are often easy from a different perspective

DRYAD

Abstract

Recent attempts to train AI models to recognize knots have produced millions of “hard” unknot diagrams resistant to simplification by Reidemeister moves, pass moves, or random walks on the Reidemeister graph. Most are easy for non-diagrammatic methods such as simplifying triangulations of the knot complement (Regina) or presentations of the knot group (SnapPy). We present ReAPR (Re-embedding And Pass Rerouting), which alternates pass-move reduction with a geometric re-embedding step. The re-embedding minimizes the total variation of a height function on the diagram subject to crossing constraints. We show that for an n-crossing diagram, the minimum total variation is 2(n–k), where k is the fewest crossings one must virtualize to make the diagram alternating; this is a combinatorial invariant of the diagram. Reprojecting the resulting embedding from a new viewpoint reveals previously hidden simplifications. ReAPR successfully simplifies all ≈2.6 million published hard-unknot examples in under 30 seconds of total CPU time. This includes a set of Kauffman's “challenge” unknots, presented as rational tangles, which appear to be surprisingly difficult for both non-diagrammatic methods.

Authors 2

1. Jason Cantarella

   University of Georgia

   Affiliation as printed

   University of Georgia

2. Clayton Shonkwiler

   Colorado State University

   Affiliation as printed

   Colorado State University