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Finite Type Invariants of W-Knotted Objects: From Alexander to Kashiwara and Vergne (self-reference), paper and related files at http://www.math.toronto.edu/~drorbn/papers/WKO/ nVassiliev [BN1] D. Bar-Natan, On the Vassiliev knot invariants, Topology 34

By Dror Bar-natan and Zsuzsanna Dancso

Abstract

Abstract. w-Knots, and more generally, w-knotted objects (w-braids, w-tangles, etc.) make a class of knotted objects which is wider but weaker than their “usual ” counterparts. To get (say) w-knots from u-knots, one has to allow non-planar “virtual ” knot diagrams, hence enlarging the the base set of knots. But then one imposes a new relation, the “overcrossings commute ” relation, further beyond the ordinary collection of Reidemeister moves, making w-knotted objects a bit weaker once again. The group of w-braids was studied (under the name “welded braids”) by Fenn, Rimanyi and Rourke FennRimanyiRourke:BraidPermutation [FRR] and was shown to be isomorphic to the McCool group McCool:BasisConjugating [Mc] of “basisconjugating” automorphisms of a free group Fn — the smallest subgroup of Aut(Fn) that contains both braids and permutations. Brendle and Hatcher BrendleHatcher:RingsAndWickets [BH], in work that traces back to Goldsmith Goldsmith:MotionGroups [Gol], have shown this group to be a group of movies of flying rings in R3

Year: 1995
OAI identifier: oai:CiteSeerX.psu:10.1.1.371.6087
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