A surgery formula for the 2-loop piece of the LMO invariant of a pair

Abstract

Let \Theta (M,K) denote the 2-loop piece of (the logarithm of) the LMO invariant of a knot K in M, a ZHS^3. Forgetting the knot (by which we mean setting diagrams with legs to zero) specialises \Theta (M,K) to \lambda (M), Casson's invariant. This note describes an extension of Casson's surgery formula for his invariant to \Theta (M,K). To be precise, we describe the effect on \Theta (M,K) of a surgery on a knot which together with K forms a boundary link in M. Whilst the presented formula does not characterise \Theta (M,K), it does allow some insight into the underlying topology.