Shadows, ribbon surfaces, and quantum invariants

Eisermann has shown that the Jones polynomial of a n-component ribbon link L⊂S3 is divided by the Jones polynomial of the trivial n-component link. We improve this theorem by extending its range of application from links in S3 to colored knotted trivalent graphs in #g(S2×S1), the connected sum of g⩾0 copies of S2×S1. We show in particular that if the Kauffman bracket of a knot in #g(S2×S1) has a pole in q=i of order n, the ribbon genus of the knot is at least n+12. We construct some families of knots in #g(S2×S1) for which this lower bound is sharp and arbitrarily big. We prove these estimates using Turaev shadows.