Quandle colourings and the rotatability of spherical knotoids up to seven crossings

Abstract

Gabrovšek and Cavicchioli have tabulated the 427 prime spherical knotoids with at most seven crossings. Their table divides them into 37 rotatable and 390 non-rotatable knotoids, but the non-rotatable column is marked as conjectural: for 28 of the seven-crossing knotoids the five polynomial invariants they computed take the same value on the knotoid and on its rotation, and an exhaustive Reidemeister search neither identified the two nor separated them. We show that all 28 are non-rotatable, using the quandle colouring counts of Gügümcü and Nelson: Fox 3-colourings separate 22 of them from their rotations, and colourings by the Alexander quandle on 𝔽₃² separate the remaining six. Eight of the 28 are also beyond the reach of Nikonov's homotopy index polynomial, the only other published invariant known to us that decides any of them. The same counts show that two entries of the rotatable column are wrong: 5₂₄ and 7₂₈₈ are not rotatable, although 7₂₈₈ is fixed by rotation composed with reversion. The reason the polynomial invariants could not decide the question is that the rotation implemented in the tabulation is the composite of mirror reflection and Turaev's symmetry, under each of which the bracket polynomial is transformed by A ↦ A⁻¹; we verify this on the published diagrams. Seven of the 390 conjecturally non-rotatable knotoids remain open. Every colouring count in this paper is checked from the published PD codes with the Lean 4 proof assistant.

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