The Jones polynomial of ribbon links

TL;DR

This paper proves divisibility properties of the Jones polynomial for ribbon links, introduces a generalized determinant with congruences, and develops new surface invariants related to Seifert surfaces and ribbon surfaces.

Contribution

It establishes divisibility and congruence properties of the Jones polynomial for ribbon links and introduces a new family of surface invariants of finite type.

Findings

Jones polynomial of ribbon links divisible by that of trivial links

Defined a generalized determinant with congruence properties

Introduced a new family of surface invariants d_k(L)

Abstract

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,...,K_n) satisfies det V(L) = det(K_1) >... det(K_n) modulo 32, whence in particular det V(L) = 1 modulo 8. These results motivate to study the power series expansion V(L) = \sum_{k=0}^\infty 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 spanning L. The invariants d_k(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.