Minimal Diagrams of Free Knots

TL;DR

This paper explores minimal diagrams of free knots, introducing a new family of diagrams from permutations that are minimal but not associated with irreducibly odd graphs, expanding understanding of free knot minimality.

Contribution

It introduces a new family of free knot diagrams derived from permutations, demonstrating minimality without the need for irreducibly odd graphs.

Findings

Identifies minimal free knot diagrams from permutation-based constructions.

Shows not all minimal diagrams are associated with irreducibly odd graphs.

Expands the class of known minimal free knot diagrams.

Abstract

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.