Quantum invariant interpretation of the a ≠ 0 truncated expressions

Determine whether, for integers m ≥ 2 and 0 ≤ a ≤ m − 2 with a ≠ 0, the truncated polynomials y^{(a)}_{m,N}(ζ_N) obtained by truncating the defining multiple-sum for the q-hypergeometric series S^{(a)}_m(q) at level N correspond to any quantum invariant (for example, Kashaev invariants of knots or links).

Background

Hikami introduced the q-hypergeometric series S^{(a)}_m(q) and conjectured that its radial limits at roots of unity are given by evaluating a truncated version of the defining series. He verified the case a = 0 by identifying both sides with the Kashaev invariant of the torus link T(2, 2m).

For a ≠ 0, Hikami noted uncertainty about whether the truncated expressions admit an interpretation as quantum invariants. The present paper proves the full radial-limit conjecture using Bailey pairs, but does not resolve the quantum-invariant interpretation for the a ≠ 0 case.