Some Computations of Ohtsuki Series

Abstract

We present some computational data on Ohtsuki series for a two parameter family of integer homology spheres obtained by surgery around what we call '2-strand knots', closures of the simplest rational tangles. This data allows us to make certain conjectures about the growth rate of the coefficients in Ohtsuki series generally, based on which we introduce an invariant which we call the slope σ(M) of a manifold M (not to be confused with slopes in hyperbolic geometry). For Seifert fibred manifolds, M , the conjectures are known to hold while π 2 σ(M) ∈ Q; furthermore if M is also an integer homology sphere, π 2 σ(M) ∈ Z. Assuming the conjectures, the numerical data enables us to give an example of a ZHS for which π 2 σ(M) ∈ Z. This paper is based on the first author's M.Sc. thesis.