Full paper in PDF:

$%M. Mulazzani, A “universal” class of 4-colored graphs, Rev. Mat. Univ. Complut. Madrid 9
(1996), no. 1, 165–195.%$

A “Universal” Class of 4-Colored Graphs

Michele MULAZZANI

Dipartimento di Matemática

Università di Bologna

Piazza di Porta San Donato 5

40127 Bologna — Italy

Università di Bologna

Piazza di Porta San Donato 5

40127 Bologna — Italy

Revised: February 27, 1995

ABSTRACT

A family of 4-colored graphs depending on three integers , , , and on a transitive pair of permutations , is constructed. Each associated topological space turns out to be a -fold branched covering of either a - or a handcuff-graph, with embedding depending on and , or a two-bridge knot or link of type . Moreover, the monodromy map is completely defined by and . In particular, when and , the space is homeomorphic to the (possibly singular) manifold , which is the branched covering of the Montesinos universal graph, associated to the pair , . This allows us to obtain a “universal” class of 4-colored graphs representing all orientable 3-dimensional singular manifolds. Further, the necessary and sufficient condition for the graph to represent a manifold is obtained and a topological interpretation of a similar construction of A. Cavicchioli is given.

1991 Mathematics Subject Classification: Primary 57M25, 57M12; Secondary 5715.