M Theory Lesson 185
For any prime $p$, M theoretic quantum information likes to specialise the knot invariants to associated roots of unity. For example, the trefoil knot at a cubed root of unity is always 1, and this normalises torus knots. This follows from the categorical $\hbar$ hierarchy, which insists that $q$ take on a fixed value determined by the categorical dimension. If this dimension were given by the number of knot crossings, as it is in Khovanov homology, it suggests a study of the numerical Jones polynomial for $q$ fixed at a primitive root of unity corresponding to the number of crossings. This is not usually done. One often encounters studies of fixed values of $q$ for all knots, but not a grading by crossing number.
A grading by strand number, however, is common in the connection between MZV algebras, knots, Feynman diagrams and chord diagrams, originally due to Kreimer but now studied by many mathematicians. The strand number is also 3 for a trefoil, or 2 for a basic braid generator associated to a qubit, so this grading is important in the analysis of quantum circuits.