Connected quandles that refute Hayashi's conjecture
Year: 2026
Type: preprint
Abstract: We construct connected quandles of orders 1890 and 5670 whose right translations $x\mapsto x*y$ have cycles of lengths 60, 84, 140 and 210. Since 60, 84 and 140 do not divide 210, this disproves the conjecture of Hayashi (2013) that in a finite connected quandle the longest cycle length of a right translation is a multiple of all the others. A quandle is a set with a self-distributive operation modeled on conjugation in a group, and quandles give invariants of knots. The conjecture was known for profiles with at most five cycles (Lages and Lopes), for orders up to 47 (Watanabe) and for affine quandles (Kajiwara and Nakayama). Filipi showed that for conjugation quandles it is equivalent to a group-theoretic conjecture, that a power of an element of a finite generating conjugacy class $C$ is central as soon as it commutes with a suitable element of $C$. Our quandles are conjugacy classes of affine maps, so they refute that conjecture too. Each cycle length is twice the product of only three of the primes 2, 3, 5 and 7, while each right translation has order 420. The construction gives an infinite family, including latin counterexamples (every left translation is also a bijection), and no counterexample from it has order below 1890. The proofs are by hand. Independent programs recompute both quandles, and one checks every quandle axiom on the full operation table. This work has not yet been reviewed by experts. The proofs, the checking programs and the text were produced by AI models (from OpenAI and Anthropic), working under the author's direction. Computer-assisted; not yet reviewed by experts. Produced by AI models under the author's direction.
Author: Graham, P.
Language: English
Topic: Mathematics and Applications
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7225337354