Volume 13, issue 2 (2009)

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The Jones polynomial of ribbon links

Michael Eisermann

Geometry & Topology 13 (2009) 623–660

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 detV (L) := [V (L)V (n)](t1), for which we derive congruences reminiscent of the Arf invariant: every ribbon link L = K1 Kn satisfies detV (L) det(K1)det(Kn) modulo 32, whence in particular detV (L) 1 modulo 8.

These results motivate to study the power series expansion V (L) = k=0dk(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.

Jones polynomial, ribbon link, slice link, nullity, signature, determinant of links
Mathematical Subject Classification 2000
Primary: 57M25
Secondary: 57M27
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
Michael Eisermann
Institut Fourier
Université Grenoble I