, Volume 328, Issue 1-2, pp 135-171
Date: 17 Oct 2003

Grope cobordism and feynman diagrams

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Abstract

We explain how the usual algebras of Feynman diagrams behave under the grope degree introduced in [CT]. We show that the Kontsevich integral rationally classifies grope cobordisms of knots in 3-space when the ‘‘class’’ is used to organize gropes. This implies that the grope cobordism equivalence relations are highly nontrivial in dimension 3. We also show that the class is not a useful organizing complexity in 4 dimensions since only the Arf invariant survives. In contrast, measuring gropes according to ‘‘height’’ does lead to very interesting 4-dimensional information [COT]. Finally, several low degree calculations are explained, in particular we show that S-equivalence is the same relation as grope cobordism based on the smallest tree with an internal vertex.

Mathematics Subject Classification (2000): 57M27
The first author was partially supported by NSF VIGRE grant DMS-9983660. The second author was partially supported by NSF grant DMS-0072775 and the Max-Planck Gesellschaft.