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(O-n) of the trivial link. This integrality property allows us to define a generalized determinant det V(L) := [V(L) / V(O-n)]((t -> -1)), for which we derive congruences reminiscent of the Arf invariant: every ribbon link L = K-1 boolean OR ... boolean OR K-n satisfies det V(L) equivalent to det(K-1) ... det(K-n) modulo 32, whence in particular det V(L) equivalent to 1 modulo 8. These results motivate to study the power series expansion V(L) = Sigma(infinity)(k=0) d(k)(L)h(k) at t= -1, instead of t = 1 as usual. We obtain a family of link invariants d(k)(L), starting with the link determinant d(0)(L) = det(L) obtained from a Seifert surface S spanniong L. The invariant d(k)(L) are not of finite type with respect to crossing change 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.