## A finite graphic calculus for 3-manifolds.(English)Zbl 0856.57009

The authors study “standard spines” of 3-manifolds. They obtain necessary and sufficient conditions for a “standard polyhedron” to be a standard spine of an orientable 3-manifold. They find a way to describe compact oriented 3-manifolds by planar graphs with some additional structure. They call these decorated graphs. The decorated graph corresponds to a standard spine with additional structure. They call these decorated spines. They go on to describe a calculus on the graphs which follows from Matveev-Piergallini moves on standard spines, these being described for example in [R. Piergallini, Standard moves for standard polyhedra and spines, Topology, 3rd Natl. Meet., Trieste/Italy 1986, Suppl. Rend. Circ. Mat. Palermo, II. Ser. 18, 391-414 (1988; Zbl 0672.57004)].

### MSC:

 57N10 Topology of general $$3$$-manifolds (MSC2010) 57M15 Relations of low-dimensional topology with graph theory

### Keywords:

spines; 3-manifolds; planar graphs

Zbl 0672.57004
Full Text:

### References:

 [1] R. Benedetti–C. Petronio,On Roberts’ proof of the Turaev-Walker theorem. To appear in Journal of Knot Theory and its Ramifications · Zbl 0890.57029 [2] B. G. Casler,An imbedding theorem for connected 3-manifolds with boundary. Proc. Amer. Math. Soc.16 (1965), 559–566 · Zbl 0129.15801 [3] R. A. Fenn–C. P. Rourke,On Kirby’s calculus on links. Topology18 (1979), 1–15 · Zbl 0413.57006 [4] R. Kirby,A calculus for framed links in S 3. Invent. Math.105 (1978), 35–46 · Zbl 0377.55001 [5] S. V. Matveev,Special spines of piecewise linear manifolds. Math. USSR-Sb.21 (1973), 279–291 · Zbl 0289.57008 [6] S. V. Matveev,Universal 3-deformations of special polyhedra. Russ. Math. Surv.42 (1987), 226–227 · Zbl 0645.57003 [7] S. V. Matveev,Transformations of special spines and the Zeeman conjecture. Math. USSR-Izv.31 (1988), 423–434 · Zbl 0676.57002 [8] C. Petronio,Standard spines and their graphic presentation as a tool for studying 3-manifolds. Tesi di Perfezionamento, Scuola Normale Superiore, Pisa, 1995 [9] R. Piergallini,Standard moves for standard polyhedra and spines. Rendiconti Circ. Mat. Palermo37, suppl. 18 (1988), 391–414 · Zbl 0672.57004 [10] C. P. Rourke–B. J. Sanderson, An introduction to piecewise linear topology. Ergebn. der Math. Bd 69, Springer-Verlag, Berlin-Heidelberg-New York, 1982 · Zbl 0477.57003 [11] V. G. Turaev–O. Ya. Viro,State sum invariants of 3-manifolds and quantum 6j-symbols. Topology31 (1992), 865–902 · Zbl 0779.57009
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