  Manturov recently introduced the idea of a free knot, i.e. an equivalence
class of virtual knots where equivalence is generated by crossing change and
virtualization moves. He showed that if a free knot diagram is associated to a
graph that is irreducibly odd, then it is minimal with respect to the number of
classical crossings. Not all minimal diagrams of free knots are associated to
irreducibly odd graphs, however. We introduce a family of free knot diagrams
that arise from certain permutations that are minimal but not irreducibly odd.

Combinatorics;Discrete Mathematics;Geometric Topology