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Finite type invariants of links and surfaces in 3-space
by
Michael Eisermann
Institut Fourier, Université Grenoble I
Slice and ribbon knots are a classical subject of knot theory ever since the seminal work of Fox and Milnor 50 years ago. Contrary to the Alexander polynomial, the Jones polynomial does not seem to reflect these topological properties. In this talk I present some results towards understanding the Jones polynomial of ribbon links, and more generally of immersed ribbon surfaces in 3-space. The right point of view is the power series expansion at t=-1 instead of t=1 as usual. The coefficients, beginning with the determinant in degree 0, are not of finite type in the sense of Vassiliev-Goussarov, but they turn out to be of finite type in the appropriate sense for (embedded or immersed) surfaces bounding links in 3-space. Motivated by this example, I shall sketch the theory of surface invariants of finite type. The aim is to reconcile quantum invariants with the classical setting of links and surfaces, and to naturally place some classical invariants of algebraic topology into the framework of an extended finite type theory.
Paper reference: arXiv:0802.2287
Date received: November 9, 2008
Copyright © 2008 by the author(s). The author(s) of this work and the organizers of the conference have granted their consent to include this abstract in Topology Atlas. Document # caxq-15.