The Jones polynomial of ribbon links

Eisermann, Michael

doi:10.2140/gt.2009.13.623

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The Jones polynomial of ribbon links

Michael Eisermann

Geometry & Topology 13 (2009) 623–660

DOI: 10.2140/gt.2009.13.623

Abstract

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

Keywords

Jones polynomial, ribbon link, slice link, nullity, signature, determinant of links

Mathematical Subject Classification 2000

Primary: 57M25

Secondary: 57M27

References

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

Volume 13, issue 2 (2009)