Free Knots and Links with Two Minimal Representatives: Explicit Examples for a Problem of Fenn, Ilyutko, Kauffman and Manturov

Problem 51 of the list of unsolved problems in virtual knot theory and combinatorial knot theory by Fenn, Ilyutko, Kauffman and Manturov (2014) asks to prove or disprove the conjecture that some free link has several minimal representatives, that is, non-isomorphic diagrams with the least number of vertices. We prove that the free knot with the Gauss word a b a e c b d f d g e g c f has exactly two minimal representatives: this diagram and the diagram a b a c e b d f d e g c g f, obtained from it by one third Reidemeister move, both with seven vertices. The same holds for the two-component free link given by the words e 1 e 3 5 2 6 4 and 1 2 3 4 5 6. Equivalence and non-isomorphism are elementary. The lower bound of seven vertices follows from Manturov's parity bracket, whose invariance we prove: each diagram has exactly one even crossing, and its bracket has two different terms, one of them an irreducible graph on the six odd crossings. That there are exactly two minimal representatives is computer-assisted; it rests on the computation of the bracket of all diagrams with seven vertices by independent programs. Kauffman and Manturov anticipated the mechanism: they observed that a diagram with a triangle is equivalent to its image under the third move, that the two can have the same unoriented state, and that minimal diagrams with triangles can be constructed; their minimality criterion, applied to a suitable explicit diagram, also answers the question. The problem list still states the conjecture. To our knowledge this is the first explicit written example with a self-contained proof; no explicit example was found in the literature accessible to us. Minimal representatives that are not related by third moves are not treated. This is an unrefereed note.