Grope Cobordism and Feynman Diagrams

Abstract

We explain how the usual algebras of Feynman diagrams behave under the grope degree introduced in "Grope cobordism of classical knots." 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 three. We also show that the class is not a useful organizing complexity in four dimensions since only the Arf invariant survives. In contrast, measuring gropes according to ``height'' does lead to very interesting four-dimensional information (Cochran-Orr-Teichner). 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.