Mock-determinant as a crossing-number bound

Determine whether the mock-determinant is an upper bound for the crossing number of every mock-alternating virtual link and every alternating virtual link.

Background

The paper defines the mock-determinant as a numerical invariant extending the classical determinant to virtual links and establishes several structural properties of this invariant. It also studies mock-alternating virtual links, a class characterized by a recursive determinant-like condition, and proves that their Jones-Kauffman polynomials are alternating.

The authors leave unresolved whether the mock-determinant controls diagrammatic complexity by bounding crossing number, both for mock-alternating virtual links and for the broader class of alternating virtual links.