# Cuts and flows of cell complexes

@article{Duval2012CutsAF,
title={Cuts and flows of cell complexes},
author={Art M. Duval and Caroline J. Klivans and Jeremy L. Martin},
journal={Journal of Algebraic Combinatorics},
year={2012},
volume={41},
pages={969-999}
}
• Published 27 June 2012
• Mathematics
• Journal of Algebraic Combinatorics
We study the vector spaces and integer lattices of cuts and flows associated with an arbitrary finite CW complex, and their relationships to group invariants including the critical group of a complex. Our results extend to higher dimension the theory of cuts and flows in graphs, most notably the work of Bacher, de la Harpe, and Nagnibeda. We construct explicit bases for the cut and flow spaces, interpret their coefficients topologically, and give sufficient conditions for them to be integral…
We study the vector spaces and integer lattices of cuts and flows of an arbitrary finite CW complex, and their relationships to its critical group and related invariants. Our results extend the
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This work considers high dimensional variants of the maximum flow and minimum cut problems in the setting of simplicial complexes and gives a combinatorial definition of a simplicial cut that seems more natural in the context of optimization problems and shows that computing such a cut is NP-hard.
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ArXiv
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The algorithmic study of some cut problems in high dimensions, namely, Topological Hitting Set and Boundary Nontrivialization, and randomized (approximation) FPT algorithms for the global variants of THS and BNT are initiated.
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• 2016
In this note we provide a higher-dimensional analogue of Tutte's celebrated theorem on colorings and flows of graphs, by showing that the theory of arithmetic Tutte polynomials and quasi-polynomials
We deduce a structurally inductive description of the determinantal probability measure associated with Kalai’s celebrated enumeration result for higher–dimensional spanning trees of the n −
. For a natural class of r × n integer matrices, we construct a non-convex polytope which periodically tiles R n . From this tiling, we provide a family of geometrically meaningful maps from a

## References

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We study the vector spaces and integer lattices of cuts and flows of an arbitrary finite CW complex, and their relationships to its critical group and related invariants. Our results extend the
• Mathematics
• 2011
We generalize the theory of critical groups from graphs to simplicial complexes. Specifically, given a simplicial complex, we define a family of abelian groups in terms of combinatorial Laplacian
Uniform spanning trees on finite graphs and their analogues on infinite graphs are a well-studied area. On a Cayley graph of a group, we show that they are related to the first $\ell^2$-Betti number
Abstract Let M be an ordered matroid and C••(M) be an exterior algebra over its underlying set E, graded by both corank and nullity. Then C•0(M) is the simplicial chain complex of IN(M), the
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A tropical curve Γ is a metric graph with possibly unbounded edges, and tropical rational functions are continuous piecewise linear functions with integer slopes. We define the complete linear system
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Let (n, k) be the class of all simplicial complexesC over a fixed set ofn vertices (2≦k≦n) such that: (1)C has a complete (k−1)-skeleton, (2)C has precisely (kn−1)k-faces, (3)Hk(C)=0. We prove that