Kawauchi’s mod-four Conway-polynomial conjecture
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For every amphicheiral knot K, does its Conway polynomial satisfy ∇K(z)≡f(z)f(−z) modulo four for some f∈Z[z]? A stronger integral identity can imply this congruence.
Why it matters and what counts as progress
Why it matters: It asks for a uniform arithmetic constraint imposed on the Conway polynomial by amphicheiral knot symmetry.
What counts as progress: Prove the modulo-four factorization for every amphicheiral knot, or find an amphicheiral Conway polynomial violating it.
How the problem developed
- 2006 · Related conjecture [1]
Jim Conant Conjectured the mod-four factorization for amphicheiral knots, independently of Kawauchi’s integral conjecture.
- 21 Jul 2026 · Proof [1]
Jim Conant, with Claude Fable 5 contributing to the initial research conversation Proved the mod-four statement as a consequence of a stronger integral statement for the remaining positive-amphicheiral case.
AI contribution
A proof of the mod 4 Kawauchi Conjecture
Conant proves that for every amphicheiral knot K, its Conway polynomial is congruent modulo 4 to f(z)f(−z) for some integer polynomial f. A stronger integral identity underlies this mod-4 conclusion.
Sources
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Publication dates and editorial changes.
Claude-suggested strategy proves the mod-4 Kawauchi conjecture