Hard unknots are often easy from a different perspective

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 least number of crossings one must virtualize to make the diagram virtually alternating; this is a combinatorial invariant of the diagram. Reprojecting the resulting embedding from a new viewpoint reveals previously hidden simplifications. ReAPR successfully simplifies every published hard-unknot example we are aware of, as well as several new collections (approximately 2.6 million examples in total) 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.