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26945 Plumbing graphs gh-baldursigurds "We implement the calculus described in [Neu1981]. The plumbing
construction creates a graph manifold (three dimensional) specified by a
plumbing graph. Each vertex of the graph specifies an S^1^-bundle over a
surface, and for each edge, the corresponding bundles are glued together
in a specific way. Each vertex is decorated by the Euler number of the
corresponding S^1^-bundle, the genus of the base of the fibration, and
the number of boundary components of the surface, r. Each edge is
decorated by +1 or -1, which specify different gluings.
The calculus of Neumann [Neu1981] consists of several operations, or
moves, which can be applied to a plumbing graph to get the same
manifold. The main result is that two graphs inducing homeomorphic (or,
equivalently, diffeomorphic) plumbed manifolds are related by a finite
sequence of these operations, or their inverses. Furthermore, each
equivalence class of graphs by these operations contains a unique
element, the minimal representative of this manifold, which can be
obtained from any representative by applying the moves while possible.
As a special case, the resolution graph of an isolated surface
singularity is a plumbing graph which describes the link of the
singularity. A plumbing graph is so obtained if and only if all genera
are positive, all edges are positive, and the associated intersection
matrix is negative definite.
Another case related to singularity theory: As proved by N\'emethi and
Szil\'ard [], the boundary of the Milnor fiber of a reduced hypersurface
singularity in complex 3-space is plumbed manifold. These do not
necessarily have negative definite plumbing graphs, and may have
negative edges as well.
We implement a class PlumbingGraph which keeps track of the graph and
the associated data. This class has as functions all the moves
introduced by Neumann in [Neu1981], as well as checking all the
minimality steps, and a function which reduces the graph to its minimal
representative (so far, the function which reduces the graph to its
minimal representative is not implemented, but it's being worked on). In
the future, we should also implement a class which is associated with
the graph a multiplicity system for a plumbing graph, as this allows for
a lot of constructions, such as computing resolution graphs for
susupension singularities, and the boundary of Milnor fibers in some
special cases.
[Neu1981] Walter D. Neumann, *A calculus for plumbing applied to the topology of complex surface singularities and degenerating complex curves*, Transactions of the American Mathematical Society, Vol. 268, No. 2 (Dec., 1981), pp. 4807--4823
" enhancement needs_review major sage-9.1 geometry plumbing graph, 3-manifolds, graph manifolds pportilla Baldur Sigurðsson N/A u/gh-baldursigurds/plumbing_graph 5f63b7385dcac9d56ec20e92144632c65e451f5c