#### Volume 13, issue 2 (2009)

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The Jones polynomial of ribbon links

### Michael Eisermann

Geometry & Topology 13 (2009) 623–660
##### 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 $detV\left(L\right):={\left[V\left(L\right)∕V\left({○}^{n}\right)\right]}_{\left(t↦-1\right)}$, 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.

##### Keywords
Jones polynomial, ribbon link, slice link, nullity, signature, determinant of links
Primary: 57M25
Secondary: 57M27
##### Publication
Received: 19 February 2008
Revised: 28 July 2008
Accepted: 27 June 2008
Preview posted: 19 November 2008
Published: 1 January 2009
Proposed: Joan Birman
Seconded: Peter Teichner, Ron Stern
##### Authors
 Michael Eisermann Institut Fourier Université Grenoble I France http://www-fourier.ujf-grenoble.fr/~eiserm