Quantum Invariants indexed by Fibered Faces of the Thurston Polytope

Abstract

We study the Gukov--Manolescu quantum invariant for oriented links. While this invariant is a single series in the knot case, we find that links admit multiple such series, each one consistent with the Melvin--Morton--Rozansky expansion of the colored Jones polynomials. We prove that convergent inverted state sums yield a family of multivariable Gukov--Manolescu series, indexed by monomials of the Alexander polynomial. We conjecture that these monomials correspond precisely to the fibered faces of the Thurston norm ball and provide extensive computational evidence for this correspondence. Further results concerning the leading term, the corresponding single-variable invariant, and the effect of partial Dehn surgery are also established.