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}
}
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… 

Cuts and Flows of Cell Complexes Art

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

Simplicial and Cellular Trees

Much information about a graph can be obtained by studying its spanning trees. On the other hand, a graph can be regarded as a 1-dimensional cell complex, raising the question of developing a theory

Simplicial and Cellular Trees Art

Much information about a graph can be obtained by studying its spanning trees. On the other hand, a graph can be regarded as a 1-dimensional cell complex, raising the question of developing a theory

Generalized max-flows and min-cuts in simplicial complexes

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.

A Simplicial Tutte "5"-flow Conjecture

This paper concerns a generalization of nowhere-zero modular q-flows from graphs to simplicial complexes of dimension d greater than 1. A modular q-flow of a simplicial complex is an element of the

The complexity of high-dimensional cuts

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.

Colorings and flows on CW complexes, Tutte quasi-polynomials and arithmetic matroids

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

Simplex links in determinantal hypertrees

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 −

A family of matrix-tree multijections

. 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

SHOWING 1-10 OF 39 REFERENCES

Cuts and Flows of Cell Complexes Art

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

Critical Groups of Simplicial Complexes

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

Random Complexes and l^2-Betti Numbers

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

The lattice of integer flows of a regular matroid

The Combinatorial Laplacian of the Tutte Complex

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

Linear systems on tropical curves

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

Computing Betti Numbers via Combinatorial Laplacians

The Laplacian and power method is used to compute Betti numbers of simplicial complexes, which has a number of advantages over other methods, both in theory and in practice, but its running time depends on a ratio, ν, of eigenvalues which the authors have yet to understand fully.

Enumeration ofQ-acyclic simplicial complexes

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

Riemann–Roch and Abel–Jacobi theory on a finite graph