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arXiv · 2405.09100

Bistellar Cluster Algebras and Piecewise Linear Invariants

Alastair Darby·Fang Li·Zhi Lu
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Abstract

Inspired by the ideas and techniques used in the study of cluster algebras we construct a new class of algebras, called bistellar cluster algebras, from closed oriented triangulated even-dimensional manifolds by performing middle-dimensional bistellar moves. This class of algebras exhibit the algebraic behaviour of middle-dimensional bistellar moves but do not satisfy the classical cluster algebra axiom: "every cluster variable in every cluster is exchangeable". Thus the construction of bistellar cluster algebras is quite different from that of a classical cluster algebra. Secondly, using bistellar cluster algebras and the techniques of combinatorial topology, we construct a direct system associated with a set of PL homeomorphic PL manifolds of dimension 2 or 4, and show that the limit of this direct system is a PL invariant.

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Alastair Darby, Fang Li, Zhi Lu. 2024-05-15. Bistellar Cluster Algebras and Piecewise Linear Invariants. https://arxiv.org/abs/2405.09100

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