The Jones polynomial of ribbon links

Abstract

For every $n$–component ribbon link $L$ we prove that the Jones polynomial $V\left(L\right)$ is divisible by the polynomial $V\left({○}^n\right)$ of the trivial link. This integrality property allows us to define a generalized determinant $detV\left(L\right):={\left[V\left(L\right)∕V\left({○}^n\right)\right]}_{\left(t\mapsto -1\right)}$, for which we derive congruences reminiscent of the Arf invariant: every ribbon link $L=K_1\cup \cdots \cup K_n$ satisfies $detV\left(L\right)\equiv det\left(K_1\right)\cdots det\left(K_n\right)$ modulo $32$, whence in particular $detV\left(L\right)\equiv 1$ modulo $8$.

These results motivate to study the power series expansion $V\left(L\right)={\sum }_{k=0}^{\infty }{d}_k\left(L\right)h^k$ at $t=-1$, instead of $t=1$ as usual. We obtain a family of link invariants $d_k\left(L\right)$, starting with the link determinant $d_0\left(L\right)=det\left(L\right)$ obtained from a Seifert surface $S$ spanning $L$. The invariants $d_k\left(L\right)$ 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.