#### Geometry & Topology Monographs, Vol. 4 (2002),

Invariants of knots and 3-manifolds (Kyoto 2001),

Paper no. 11, pages 161--181.

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

### Andrew Kricker

**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.
**Keywords**.
Casson's invariant, LMO invariant, boundary link, surgery

**AMS subject classification**.
Primary: 57M27.
Secondary: 57M25.

**E-print:** `arXiv:math.GT/0211057`

Submitted to GT on 19 December 2001.
(Revised 6 August 2002.)
Paper accepted 10 September 2002.
Paper published 21 September 2002.

Notes on file formats
Andrew Kricker

Department of Mathematics, University of Toronto

Ontario, M5S 1A1, Canada

Email: akricker@math.toronto.edu

GTM home page

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