Kirchhoff’s theorems in higher dimensions and Reidemeister torsion

Catanzaro, Michael J.; Chernyak, Vladimir; Klein, John R.

doi:10.4310/hha.2015.v17.n1.a8

Cited by 22 publications

(40 citation statements)

References 5 publications

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“…Kalai's subcomplexes are higher dimensional analogs of spanning trees, since when k ¼ 1, they are just the spanning trees of the 1-skeleton of D n . Subsequent work in this area extended Kalai's result to more general complexes (Catanzaro et al 2015;Cappell and Miller 2015;Duval et al 2009Duval et al , 2011Duval et al , 2015Lyons 2009). With respect to these efforts, the notion of spanning tree was extended to higher dimensions (cf.…”

Section: Improved Higher Matrix-tree Theoremsmentioning

confidence: 84%

“…one gets another proof of our higher dimensional analog of Kirchhoff's theorem on electrical networks (Catanzaro et al 2015) (see Remark 3.6 below).…”

Section: Remark 116mentioning

confidence: 94%

“…The latter are certain subcomplexes of X of dimension d À 1 which are a higher dimensional analog of the vertices of a graph. They are homologically dual to the higher dimensional spanning trees of Catanzaro et al (2015) and hence their name. In the following definition, b k ðXÞ ¼ dim H k ðX; QÞ denotes the kth Betti number of X, and X ðkÞ denotes the k-skeleton of X.…”

Section: The Combinatorial Hodge Problemmentioning

confidence: 99%

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A higher Boltzmann distribution

Catanzaro

Chernyak

Klein

2017

J Appl. and Comput. Topology

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We characterize the classical Boltzmann distribution as the unique solution to a combinatorial Hodge theory problem in homological degree zero on a finite graph. By substituting for the graph a CW complex X and a choice of degree d dim X, we define by direct analogy a higher dimensional Boltzmann distribution q B as a certain real-valued cellular ðd À 1Þ-cycle. We then give an explicit formula for q B . We explain how these ideas relate to the Higher Kirchhoff Network Theorem of Catanzaro et al. (Homol Homotopy Appl 17:165-189, 2015). We also deduce an improved version of the Higher Matrix-Tree Theorems of Catanzaro et al. (Homol Homotopy Appl 17:165-189, 2015).

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Section: Improved Higher Matrix-tree Theoremsmentioning

confidence: 84%

“…one gets another proof of our higher dimensional analog of Kirchhoff's theorem on electrical networks (Catanzaro et al 2015) (see Remark 3.6 below).…”

Section: Remark 116mentioning

confidence: 94%

Section: The Combinatorial Hodge Problemmentioning

confidence: 99%

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A higher Boltzmann distribution

Catanzaro

Chernyak

Klein

2017

J Appl. and Comput. Topology

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“…This step follows, mutatis mutandis, by the proof of [CCK,proposition 4.2]. We emphasize that γ is independent of R.…”

Section: Proof Of Theorem Cmentioning

confidence: 97%

“…We work perturbatively, following a modified version of [CCK,proposition 5.2]. To this end, let β ∈ R + be the perturbation parameter and fix a ρ-spanning tree T .…”

Section: Proof Of Theorem Cmentioning

confidence: 99%

On Kirchhoff’s theorems with coefficients in a line bundle

Catanzaro

Chernyak

Klein

2013

Homology, Homotopy and Applications

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Simplicial and cellular trees

Duval

Klivans

Martin

2016

The IMA Volumes in Mathematics and Its Applications

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How can redundancy be eliminated from a network, and what can be said about the resulting substructures? In graph-theoretic terms, these substructures are spanning trees. The collection of all spanning trees of a graph has the structure of a matroid basis system; this observation connects trees to algebraic combinatorics and explains why many graph algorithms can be made computationally efficient. The number of spanning trees of a graph measures its complexity as a network, and there are classical and efficient linear-algebraic tools for calculating this number, as well as for enumerating trees more finely. The algebra of trees is key in studying dynamical systems on graphs, notably the abelian sandpile model (known in other forms as the chip-firing game or dollar game), whose possible states are encoded by a group of size equal to the complexity of the underlying graph.This chapter is about the more recent theory of spanning trees in cell complexes, which are natural higher-dimensional analogues of graphs. The theory for cell complexes parallels the graph-theoretical version in many ways, including the connection to matroid theory. However, higher-dimensional spaces can have much richer topology, which complicates the algebraic and enumerative parts of the story in a very concrete way. It turns out that the simple number of spanning trees of a cell complex is not a well-behaved invariant, and it is better to account for topological complexity in the enumeration. On the other hand, many essential algebraic tools for working with spanning trees of a graph do extend well to arbitrary dimension.

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Kirchhoff’s theorems in higher dimensions and Reidemeister torsion

Cited by 22 publications

References 5 publications

A higher Boltzmann distribution

A higher Boltzmann distribution

On Kirchhoff’s theorems with coefficients in a line bundle

Simplicial and cellular trees

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