Finite type invariants of 3-manifolds

Abstract

A theory of finite type invariants for arbitrary compact oriented 3-manifolds is proposed, and illustrated through many examples arising from both classical and quantum topology. The theory is seen to be highly non-trivial even for manifolds with large first betti number, encompassing much of the complexity of Ohtsuki's theory for homology spheres. (For example, it is seen that the quantum SO(3) invariants, though not of finite type, are determined by finite type invariants.) The algebraic structure of the set of all finite type invariants is investigated, along with a combinatorial model for the theory in terms of trivalent "Feynman diagrams".

Final version for publication, with figures. The most significant changes from the original posted version are in the exposition of section 3 (on the Conway polynomial) and section 4 (on quantum invariants)

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• Algebraic linking numbers of knots in 3-manifolds
• Cohomology rings, Rochlin function, linking pairing and the Goussarov--Habiro theory of three--manifolds
• Kirby calculus for null-homologous framed links in 3-manifolds
• Finite type invariants of knots in homology 3-spheres with respect to null LP-surgeries