On the Frohman Kania-bartoszynska Ideal

The Frohman Kania-Bartoszynska ideal is an invariant associated to a 3-manifold with boundary and a prime p ≥ 5. We give some estimates of this ideal. We also calculate this invariant for some 3-manifolds constructed by doing surgery on a knot in the complement of another knot. Let p = 2d+ 1 ≥ 5 be a prime and let O = { Z[ζp] if p ≡ −1 (mod 4) , Z[ζp, i] = Z[ζ4p] if p ≡ 1 (mod 4) . If M is an oriented connected closed 3-manifold, define Ip(M) = D 〈(M)〉p, where 〈 〉p is the invariant defined in [BHMV], D = 〈S3〉−1 p and M is given a p1-structure with σ-invariant zero. Alternatively M is given the weight zero [G]. This is the same normalization as in [MR]. For example, Ip(S ) = 1. The invariant Ip takes values in O by [M, MR]. Here is the definition of the Frohman Kania-Bartoszynska ideal invariant, as given in [GM]. Definition 1 ([FK]). Given a connected 3-manifold N with boundary, let Jp(N) be the ideal in O generated by {Ip(M)|M is a closed connected oriented 3-manifold containing N}. In the case p ≡ 1 (mod 4), we define J+ p (N) = Jp(N) ∩ Z[ζp]. Frohman and Kania-Bartoszynska actually made this definition using the SU(2) theory in place of the SO(3) theory used here. If we replace Ip in the above definition with I2p (in the sense of [MR]) and replace Z[ζp] with Z[ζ8p], then we obtain the ideal invariant of [FK] exactly. They gave some interesting examples of punctured 3-manifolds that could not embed in other 3-manifolds, even though this is not ruled out by homology or homotopy type. We remark that results for punctured 3-manifolds follow immediately from the fact that quantum invariants of closed 3-manifolds are multiplicative under connect sum and do not require the structure of a TQFT. They were able to show that the ideal invariant, associated to a certain Turaev-Viro invariant, was non-trivial for the union of two solid tori glued together by identifying neighborhoods of (2, 1) curves on their boundary. This is a 3-manifold with boundary a torus and first homology Z ⊕ Z2. We also recover this estimate in Remark 16. We note that if p ≡ 1 (mod 4), the ideal Jp(N) is generated by scalars, according to Definition 1, which are either in Z[ζp] or in iZ[ζp]. Thus Jp(N) is generated over O by J+ p (N). Thus J + p (N) contains the same information as Jp(N). We make the following definition so that the ideal we study lies in Z[ζp] in every case. This is convenient as Z[ζp] has a simpler ideal structure than Z[i, ζp]. Date: November 15, 2004. This research was partially supported by NSF-DMS-0203486.