Abstract
We introduce a representation of compact 3-manifolds without spherical boundary components via (regular) 4-colored graphs, which turns out to be very convenient for computer aided study and tabulation. Our construction is a direct generalization of the one given in the 1980s by S. Lins for closed 3-manifolds, which is in turn dual to the earlier construction introduced by Pezzana’s school in Modena. In this context we establish some results concerning fundamental groups, connected sums, moves between graphs representing the same manifold, Heegaard genus and complexity, as well as an enumeration and classification of compact 3-manifolds representable by graphs with few vertices ( $${\le }6$$ in the non-orientable case and $${\le }8$$ in the orientable one).
Highlights
Introduction and preliminariesThe representation of closed 3-manifolds by 4-colored graphs has been independently introduced by S
In this paper we show that any 4-colored graph, with no additional conditions, can represent a compact 3-manifold without spherical boundary components, and the whole class of such manifolds admits a representation of this type
In the non-orientable case, we start with the solid Klein bottle H1, which can be represented by the 4-colored graph Γ′1 of Figure 1: it has 6 vertices a, b, c, A, B, C, the same c-edges (c ∈ {0, 1, 3}) as Γ1, and the 2-edges connecting b with c, A with C and a with B
Summary
The representation of closed 3-manifolds by 4-colored graphs has been independently introduced by S. A compression body Kg of genus g is a 3-manifold with boundary obtained from Sg × I, where I = [0, 1], by attaching a finite set of 2-handles along a system of curves (called attaching circles) on Sg × {0} and filling in with balls all the spherical boundary components of the resulting manifold, except for Sg × {1} when g = 0. The triple (Sg, K′, K′′) is called a generalized Heegaard splitting of genus g of M It is a well-known fact that each compact connected 3-manifold without spherical boundary components admits a Heegaard splitting, and at least one of the two compression bodies can be assumed to be a handlebody (in this case the splitting is called Heegaard splitting). For general PL-topology and elementary notions about graphs and embeddings, we refer to [27] and [36] respectively
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