Abstract
A $d$-dimensional simplicial complex is balanced if the underlying graph is $(d+1)$-colorable. We present an implementation of cross-flips, a set of local moves introduced by Izmestiev, Klee and Novik which connect any two PL-homeomorphic balanced combinatorial manifolds. As a result we exhibit a vertex minimal balanced triangulation of the real projective plane, of the dunce hat and of the real projective space, as well as several balanced triangulations of surfaces and 3-manifolds on few vertices. In particular we construct small balanced triangulations of the 3-sphere that are non-shellable and shellable but not vertex decomposable.
Highlights
The study of the number of faces in each dimension that a triangulation of a manifold M can have is a very classical and hard problem in combinatorial topology
In this article we focus on the family of balanced simplicial complexes, i.e., d-dimensional complexes whose underlying graph is (d + 1)-colorable, in the classical graph theoretic sense
In particular we can ask the following question: what is the minimum number of vertices that a balanced triangulation of a manifold M can have? Izmestiev, Klee and Novik introduced a finite set of local moves called cross-flips, which preserves balancedness, the PL-homeomorphism type, and suffices to connect any two balanced combinatorial triangulations of a manifold
Summary
The study of the number of faces in each dimension that a triangulation of a manifold M can have is a very classical and hard problem in combinatorial topology. [3] designed a computer program called BISTELLAR, employing bistellar flips in order to obtain triangulations on few vertices and heuristically recognize the homeomorphism type. This tool led to a significant number of small or even vertex-minimal triangulations which are listed in The Manifold Page [14], along with many other interesting examples (see [5] and [3]). Klee and Novik introduced a finite set of local moves called cross-flips, which preserves balancedness, the PL-homeomorphism type, and suffices to connect any two balanced combinatorial triangulations of a manifold. The source code and the list of facets of all simplicial complexes appearing in this paper are made available in [21]
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
