Grope cobordism of classical knots

Details Grope cobordism of classical knots Grope cobordism of classical knots

2000-12-14

arxiv.org/abs/math/0012118v3

Topology 43 (2004), no. 1, 119--156

Mathematics

Geometric Topology

Final version. To appear in Topology. See http://www.math.cornell.edu/~jconant/pagethree.html for a PDF file with better fig

Scientific paper

We explain the notion of a grope cobordism between two knots in a 3-manifold. Each grope cobordism has a type that can be described by a rooted unitrivalent tree. By filtering these trees in different ways, we show how the Goussarov-Habiro approach to finite type invariants of knots is closely related to our notion of grope cobordism. Thus our results can be viewed as a geometric interpretation of finite type invariants. An interesting refinement we study are knots modulo symmetric grope cobordism in 3-space. On one hand this theory maps onto the usual Vassiliev theory and on the other hand it maps onto the Cochran-Orr-Teichner filtration of the knot concordance group, via symmetric grope cobordism in 4-space. In particular, the graded theory contains information on finite type invariants (with degree h terms mapping to Vassiliev degree 2^h), Blanchfield forms or S-equivalence at h=2, Casson-Gordon invariants at h=3, and for h=4 one has the new von Neumann signatures of a knot.

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