Mathematische Annalen

, Volume 328, Issue 1, pp 135–171

Grope cobordism and feynman diagrams

Article

DOI: 10.1007/s00208-003-0477-y

Cite this article as:
Conant, J. & Teichner, P. Math. Ann. (2004) 328: 135. doi:10.1007/s00208-003-0477-y

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.

Key words or phrases

Grope cobordismFeynman diagramsVassiliev invariants

Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  1. 1.Department of MathematicsCornell UniversityIthacaUSA
  2. 2.Department of MathematicsUniversity of California in San Diego