Acyclonestohedra in Oriented Matroid Theory

• Acyclonestohedra are convex polytopes associated with acyclic nested complexes on oriented matroids, capturing key combinatorial and geometric properties.
• The construction employs evaluation space sections of nestohedra with carefully chosen coefficients to isolate and realize only the acyclic faces.
• This framework unifies and generalizes associahedra, graph associahedra, and poset associahedra by providing explicit coordinate descriptions and links to compactification theory.

Acyclonestohedra are polytopes associated with acyclic nested complexes on oriented matroids. In the realizable case, an acyclonestohedron is obtained by intersecting a nestohedron with the evaluation space of a vector configuration, and the resulting section selects exactly the acyclic part of the corresponding boolean nested complex; equivalently, the acyclic nested complex is the boundary complex of the polar of that section (Mantovani et al., 19 Sep 2025). In a parallel formulation, acyclonestohedra are convex polytopes whose faces are indexed by acyclic nestings on an oriented building set $(S,\mathcal B,\mathcal C)$, and they generalize Stasheff associahedra, graph associahedra, nestohedra, and poset associahedra (Forcey et al., 13 Jul 2025).

1. Lattice-theoretic and oriented-matroid foundations

A building set on a finite lattice $L$ is a subset $B\subseteq L_{>0}$ such that every interval below an element $Y$ factors as a product of the intervals below the maximal blocks of $B$ contained in $Y$ (Mantovani et al., 19 Sep 2025). In the boolean case $L=2^S$, this reduces to the condition that $B$ contains all singletons and is closed under unions of intersecting members. Given a building set $B$, a nested set is a subset $\mathcal N\subseteq B$ containing the connected components $L$0 such that any collection of pairwise incomparable members has join not in $L$1. The corresponding nested complex $L$2 is the simplicial complex of these nested sets, with the connected components removed when passing to the simplicial complex.

For boolean building sets, nested complexes are realized by nestohedra

$L$3

where $L$4 and $L$5 for $L$6 (Mantovani et al., 19 Sep 2025). Their normal fan is the nested fan, and the boundary complex of the polar is isomorphic to the nested complex.

The oriented-matroid generalization replaces the boolean lattice by the Las Vergnas face lattice $L$7 of an oriented matroid $L$8. A subset $L$9 is a face precisely when

$B\subseteq L_{>0}$0

If $B\subseteq L_{>0}$1 is realizable by a vector configuration $B\subseteq L_{>0}$2 and is acyclic, then this face lattice agrees with the face lattice of the cone $B\subseteq L_{>0}$3, and therefore with the face lattice of a polytope obtained by slicing that cone (Mantovani et al., 19 Sep 2025). A facial building set is a building set on this face lattice, and the associated nested complex is the facial nested complex.

The abstract of "Facial nested complexes and acyclonestohedra" states that nested complexes of building sets on Las Vergnas face lattices are obtained by iterated stellar subdivisions of the positive tope, and that in the realizable case the nested complex is isomorphic to the boundary complex of a polytope (Mantovani et al., 19 Sep 2025). This places acyclonestohedra in the intersection of nestohedral combinatorics and oriented-matroid face theory.

2. Acyclic nested complexes and the facial–acyclic equivalence

The paper introduces an acyclic building-set viewpoint by considering an oriented building set $B\subseteq L_{>0}$4, meaning a building set $B\subseteq L_{>0}$5 on the same ground set as $B\subseteq L_{>0}$6 such that the support of every circuit of $B\subseteq L_{>0}$7 lies in $B\subseteq L_{>0}$8 (Mantovani et al., 19 Sep 2025). A nested set $B\subseteq L_{>0}$9 on $Y$0 is called acyclic if, after restricting and contracting along every partial union inside $Y$1, the resulting oriented matroid remains acyclic.

Several equivalent criteria are given. For a nested set $Y$2, acyclicity is equivalent to the condition that for every subcollection $Y$3, the union $Y$4 is a face of $Y$5; equivalently, no circuit has all its negative part inside such a union while some positive part escapes it (Mantovani et al., 19 Sep 2025). The acyclic nested complex $Y$6 is the simplicial complex whose faces are the acyclic nested sets.

A central theorem identifies the facial and acyclic constructions. If $Y$7 is the facial part of $Y$8, then

$Y$9

The same theorem is accompanied by a structural statement: the map $B$0 sends oriented building sets onto facial building sets, and any facial building set can be extended to an oriented one by adjoining circuit supports and taking building closure (Mantovani et al., 19 Sep 2025).

This equivalence is the combinatorial core of the subject. It shows that acyclonestohedra do not introduce an unrelated class of complexes, but rather isolate the acyclic subcomplex already latent in a suitably chosen boolean nested complex. A plausible implication is that the combinatorics of acyclonestohedra is best viewed as a face-lattice refinement problem controlled by oriented-matroid acyclicity.

3. Boolean embeddings, evaluation spaces, and the section construction

The geometric realization proceeds through embeddings of nested complexes into boolean nested complexes. More generally, if $B$1 is an order embedding satisfying certain tameness conditions, then nested complexes on $B$2 can embed into nested complexes on $B$3 (Mantovani et al., 19 Sep 2025). A map is tame if it is an order embedding and is either atom-exhaustive, or join-preserving, or cover-preserving. Under these hypotheses, if $B$4 is the preimage of an $B$5-building set $B$6, then $B$7 is $B$8-compatible, so the $B$9-nested complex embeds as a subcomplex of the $Y$0-nested complex.

For finite atomic lattices there is a canonical embedding

$Y$1

where $Y$2 is the set of atoms. This embedding is atom-exhaustive and tame, and every building set is pushable across it. Hence every nested complex on a finite atomic lattice embeds into a boolean nested complex (Mantovani et al., 19 Sep 2025). In the oriented-matroid setting, this yields the key interpretation: the facial nested complex, equivalently the acyclic nested complex, sits inside the boolean nested complex of a suitable boolean building set, and the acyclic faces are precisely those compatible with the oriented matroid.

Let $Y$3 be a realizable oriented matroid. Its evaluation space $Y$4 is the subspace spanned by all evaluation vectors $Y$5, equivalently

$Y$6

Choosing coefficients $Y$7 with

$Y$8

for $Y$9 sufficiently large, the acyclonestohedron is defined by the section

$L=2^S$0

(Mantovani et al., 19 Sep 2025).

The role of the large coefficients is explicit. If a nested set is not acyclic, then the corresponding face of the nestohedron lies entirely on one side of some circuit hyperplane and misses $L=2^S$1; if it is acyclic, then its face intersects $L=2^S$2. The main realization theorem is

$L=2^S$3

Thus the section realizes exactly the acyclic nested complex, and no extraneous boolean faces survive (Mantovani et al., 19 Sep 2025).

There is also an equivalent realization in the ambient space $L=2^S$4, described by inequalities

$L=2^S$5

with equalities for the connected components of $L=2^S$6. The resulting polytope is affinely equivalent to $L=2^S$7 (Mantovani et al., 19 Sep 2025). This provides explicit coordinates in the natural realization space of the matroid rather than only an abstract section description.

4. Special cases, recoveries, and unified families

Acyclonestohedra recover several previously studied families exactly. Poset associahedra are the graphical acyclonestohedra: for a poset $L=2^S$8, Galashin’s piping complex is identified with the acyclic nested complex of a graphical oriented building set built from the Hasse diagram of $L=2^S$9 (Mantovani et al., 19 Sep 2025). In the graphical case, the acyclonestohedron becomes a graph associahedron sectioned by the evaluation space of the graph’s oriented matroid. This recovers Galashin’s polytopality results, provides explicit coordinates, and answers some of his open questions.

The same framework extends to affine poset cyclohedra, again as acyclic nested complexes for a suitable affine oriented matroid, with explicit section-realizations (Mantovani et al., 19 Sep 2025). More broadly, the paper states that the framework subsumes ordinary nestohedra, hyperoctahedral nestohedra, design graph associahedra, and permutopermutohedra, together with graphical and affine graphical constructions. The abstract emphasizes that poset associahedra are the graphical acyclonestohedra and that the theory generalizes the poset associahedra recently introduced by P. Galashin, from order polytopes to any polytope (Mantovani et al., 19 Sep 2025).

Several precise degeneration statements clarify the range of the construction. If $B$0 is linearly independent, then $B$1, so the acyclonestohedron is just the original nestohedron. For an oriented forest, the graphical acyclonestohedron is the usual graph associahedron. For a poset whose Hasse diagram is a tree, one recovers the associahedron or permutahedron depending on the tree shape (Mantovani et al., 19 Sep 2025). The paper also notes that some truncations are redundant: facets corresponding to blocks not visible in the evaluation space never meet the section. This explains why the section model is more efficient than a naive iterated truncation.

The framework also interacts with compactification theory. By a theorem of Gaiffi, the interior of any polytope admits a stratified $B$2 compactification whose strata are indexed by the faces of the relevant facial nested complex; in the realizable case, the acyclonestohedron provides the combinatorial model for such compactifications. The paper further connects this boundary structure to wondertopes of Brauner–Eur–Pratt–Vlad (Mantovani et al., 19 Sep 2025).

5. Oriented building sets, canonical forms, and acyclonesto-cosmohedra

A second line of development treats acyclonestohedra as positive geometries. "Acyclonesto-cosmohedra" defines them as convex polytopes whose face poset is the poset of acyclic nestings of an oriented building set $B$3, ordered by reverse inclusion (Forcey et al., 13 Jul 2025). Here $B$4 is a building set on $B$5, and $B$6 is an acyclic realizable matroid on the same ground set. If $B$7, the acyclonestohedron reduces to the ordinary nestohedron or graph associahedron. For a totally ordered set, the building set is the family of intervals

$B$8

and the corresponding acyclonestohedron is exactly the Stasheff associahedron (Forcey et al., 13 Jul 2025).

The acyclicity condition is imposed locally along a nesting. An acyclic nesting $B$9 is a nesting such that for every $B$0, the matroid obtained by restricting to $B$1 and contracting by the union of smaller nests is acyclic. If the oriented matroid is realizable by vectors $B$2, then the polytope has dimension

$B$3

where the vectors span a $B$4-dimensional space (Forcey et al., 13 Jul 2025).

The paper assigns affine kinematic variables

$B$5

with cut parameters $B$6, and realizes the polytope by the inequalities

$B$7

for allowed $B$8, together with

$B$9

When the $\mathcal N\subseteq B$0 are linearly dependent, additional constraints on the $\mathcal N\subseteq B$1 appear; a sufficient hierarchy given in the paper is

$\mathcal N\subseteq B$2

(Forcey et al., 13 Jul 2025).

As a positive geometry, the acyclonestohedron has canonical form

$\mathcal N\subseteq B$3

where the rational function

$\mathcal N\subseteq B$4

is the amplitube, summed over maximal acyclic tubings $\mathcal N\subseteq B$5 (Forcey et al., 13 Jul 2025). The factorization property is explicit: a facet corresponding to $\mathcal N\subseteq B$6 factorizes into smaller acyclonestohedra obtained by restriction and contraction, and residues on poles factorize into products of lower-dimensional amplitubes. The paper interprets this as the geometric realization of locality and unitarity.

The same article introduces acyclonesto-cosmohedra, obtained by a further truncation in which faces are labeled by nested nestings $\mathcal N\subseteq B$7. A face has codimension

$\mathcal N\subseteq B$8

including the improper nest, so $\mathcal N\subseteq B$9, and these polytopes are generally non-simple for $L$00 (Forcey et al., 13 Jul 2025). They are realized by variables

$L$01

with inequalities

$L$02

and a hierarchy $L$03, $L$04 whenever $L$05 (Forcey et al., 13 Jul 2025).

Their canonical forms define a cosmological amplitube

$L$06

interpreted as a generalization of cosmological wavefunction coefficients. The paper also gives evidence that acyclonesto-cosmohedra can be obtained as sections of graph cosmohedra; for the diamond and bowtie posets, the corresponding acyclonesto-cosmohedra are polygons with $L$07 and $L$08 sides respectively, and both are realized as sections of the graph cosmohedron of $L$09 (Forcey et al., 13 Jul 2025).

6. Relation to associahedra, cyclohedra, and acyclotopes

The term acyclonestohedron does not appear in "Associahedra, cyclohedra and inversion of power series" (Aguiar et al., 2020). That paper nevertheless studies a nearby polyhedral-combinatorial framework: the Hopf monoid $L$10 of sets of cycles and paths, with paths corresponding to associahedra and cycles corresponding to cyclohedra. For a graph $L$11, the graph associahedron $L$12 is used throughout, and its faces are in order-reversing bijection with tubings $L$13, with

$L$14

(Aguiar et al., 2020). The paper gives cancellation-free and grouping-free antipode formulas, describes the character group $L$15 as pairs of power series, and expresses inversion in terms of faces of associahedra and cyclohedra. This suggests a precursor to any framework intended to unify acyclic path-type and cyclic cycle-type nested structures.

A separate possible source of confusion is the term acyclotope. "Acyclotopes and Tocyclotopes" states that the term “Acyclonestohedra” does not appear there, and that the closest and essentially intended object is the acyclotope, namely the graphical zonotope $L$16, whose vertices correspond to acyclic orientations of a graph $L$17 (Bach et al., 2024). That paper then introduces the dual tocyclotope, whose vertices correspond to totally cyclic orientations, and develops Ehrhart formulas for graphs, signed graphs, and general integer matrices. The acyclotope/tocyclotope theory is therefore a zonotopal and matroid-dual construction, whereas acyclonestohedra arise from nested complexes, oriented building sets, and evaluation-space sections of nestohedra. The two theories are related by their use of acyclic and cyclic combinatorics, but they denote different polyhedral objects.

Within this broader landscape, acyclonestohedra occupy the nestohedral/oriented-matroid side of the subject. Their defining feature is that acyclicity is enforced by the oriented matroid and is then realized geometrically as a section selecting precisely the admissible nested faces. That feature distinguishes them both from classical graph associahedra, where all nested faces are present, and from acyclotopes, where the basic polytope is a zonotope encoding orientations rather than a section of a nestohedron.