A finite graphic calculus for 3-manifolds

Abstract

In this paper we provide a presentation for compact oriented 3-manifolds with non-empty boundary up to orientation-preserving homeomorphism via a calculus on suitable finite planar graphs with extra structure (decorated graphs). Closed manifolds are included in this representation by removing a 3-ball. Decorated graphs have an intrinsic geometric counterpart, as they are actually obtained by considering standard spines of the manifold and extra structure on them (decorated spines). The calculus on graphs is derived from the Matveev-Piergallini moves on standard spines ([2], [6], [7], [9]) which we re-examine and adapt to our setting (in particular in Section 1 we establish an oriented theory). A comparison with the presentation of closed 3-manifolds via surgery on framed links in S 3 and the corresponding Kirby calculus [3], [4] allows to single out peculiarities of our graphic calculus. If one represents links by generic projections there are formal analogies between the two presentations, in particular both are supported by quadrivalent planar graphs with simple normal crossings and both express the change of orientation on manifolds by a simple involution on the set of graphs. The main differences are that edge-colours are taken in Z for framed links and in Z3 for us, and that our calculus is generated by a finite number (in a strict sense) of local moves, while on one hand the band move of Kirby calculus is not local, and on the other hand the local general Kirby move depends on the arbitrarily large number of strands involved. The finiteness of our calculus can be exploited to construct polynomial invariants of spines and 3-manifolds in a way formally very close to the elementary definition of the Kauffman bracket invariant of framed links. For every choice of initial data (for which there is a wide freedom) the construction produces an ideal in a polynomial ring, explicitly given by a finite set of integral generators, and a process which to every decorated graph associates a polynomial. This polynomial is defined as a state sum which satisfies certain linear skein relations, and the class of the polynomial modulo the ideal is invariant under the calculus. This construction is widely discussed in [8], and non-triviality of the invariants produced is supported by the proof that Turaev-Viro invariants [11] appear in this framework, with a very simple choice of initial data. Our calculus has also been used in [1] for a formal algebraic treatment of Roberts' approach to the Turner-Walker theorem. For the reader's convenience we state the main result which we will establish. We confine ourselves here to the case of oriented manifolds, because the presentation is particularly easy to describe in this case, but a similar graphic presentation is provided below also for non-oriented manifolds. We start by introducing a few definitions. Consider a finite planar quadrivalent graph F with simple normal crossings, and assume that some vertices of F are marked. In the sequel given such a graph we will always call edges of F those obtained by ignoring the vertices which are not marked (in other words, edges are locally embedded segments with marked endpoints). Remark that edges might well not cover the graph.