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
Received: May 13, 1994
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.