The Jones polynomial of ribbon links
Abstract
For every n-component ribbon link L we prove that the Jones polynomial V(L) is divisible by the polynomial V(On) of the trivial link. This integrality property allows us to define a generalized determinant det V(L) := [V(L)/V(On)](t=-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) = Σk=0∞ dk(L) hk 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.