Maximal conjecture on free parameters in HOMFLY polynomials of virtual knots

A dessin is a graph-like object used in the hypercube construction of the HOMFLY polynomial, and its $q$-dimension is the parameter assigned to it. The HOMFLY polynomial for a virtual knot is required to remain topologically invariant. The notation $[N-1]$, $[N-2]$, $[2]$, $1$, and $[N-3]$ denotes the corresponding decomposition rules for dessins; consider dessins that cannot be decomposed using any of these rules. Factorization of composites can be imposed as an additional requirement.

Maximal conjecture. An infinite number of free parameters, namely values of $q$-dimensions for dessins that cannot be decomposed using the $[N-1]$-, $[N-2]$-, $[2]$-, $1$-, and $[N-3]$-rules, can be put into the HOMFLY polynomial for virtual knots while preserving its topological invariance. Many of these parameters are consistent with, that is, not fixed by, factorization of composites.

This is the stronger companion to the minimal conjecture and addresses the persistence of ambiguity after the standard decomposition rules and composite-factorization requirement are taken into account. It is presented as an empirically checked conjecture; the source reports checks for planar diagrams with up to and including seven non-virtual crossings and arbitrarily many virtual crossings.