The Jones polynomial of ribbon links

For every n–component ribbon link L we prove that the Jones polynomial V(L) is divisible by the polynomial V(©n) of the trivial link. This integrality property allows us to define a generalized determinant det V(L) := [V(L)/V(©)](t 7→−1) , for which we derive congruences reminiscent of the Arf invariant: every ribbon link L = K1∪· · ·∪Kn satisfies det V(L) ≡ det(K1) · · · det(Kn) modulo 32, whence in particular det V(L) ≡ 1 modulo 8. These results motivate to study the power series expansion V(L) = ∑∞ k=0 dk(L)h k at t = −1, instead of t = 1 as usual. We obtain a family of link invariants dk(L), starting with the link determinant d0(L) = det(L) obtained from a Seifert surface S spanning L . The invariants dk(L) are not of finite type with respect to crossing changes of L , but they turn out to be of finite type with respect to band crossing changes of S . This discovery is the starting point of a theory of surface invariants of finite type, which promises to reconcile quantum invariants with the theory of Seifert surfaces, or more generally ribbon surfaces.