Miller–Wong conjecture on the odd invariant of smooth 2-knots

Let FF be a smooth 22-knot, let n(F)n(F) denote its odd Khovanov invariant, and let Σ(F)\Sigma(F) denote the branched double-cover associated to FF. Miller–Wong's conjecture. For every smooth 22-knot,

n(F)=H1(Σ(F);Z).n(F)=\lvert H_1(\Sigma(F);\mathbb{Z})\rvert.

The conjecture generalizes the theorem for ribbon 22-knots stated earlier in the paper. More recently, Spyropoulos, Vidyarthi, and Zhang proved the conjecture in general, so it is solved.

Sources & referencesView supporting material

Primary source

Jacob Migdail and Stephan Wehrli, “A module structure on odd Khovanov homology and the odd invariant for ribbon 2-knots”, arXiv:2607.04018 (2026).

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