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Melissa Sherman-Bennett

Publications and source records attributed to Melissa Sherman-Bennett.

At least 19 recordsLinked to original sources
arXiv2026

A dimer view on Fox's trapezoidal conjecture

Fox's conjecture (1962) states that the sequence of absolute values of the coefficients of the Alexander polynomial of alternating links is trapezoidal. While the conjecture remains open in general, a number of special cases have been settled, some quite recently: Fox's conjecture was shown to hold for special alternating links by Hafner, Mészáros, and Vidinas (2023) and for certain diagrammatic Murasugi sums of special alternating links by Azarpendar, Juhász, and Kálmán (2024). In this paper, we give an alternative proof of Azarpendar, Juhász, and Kálmán's aforementioned beautiful result via a dimer model for the Alexander polynomial. In doing so, we not only obtain a significantly shorter proof of Azarpendar, Juhász, and Kálmán's result than the original, but we also obtain several theorems of independent interest regarding the Alexander polynomial, which are readily visible from the dimer point of view.

math.CO↗
arXiv2026

Cluster algebras and tilings for the m=4 amplituhedron

The amplituhedron $A_{n,k,m}(Z)$ is the image of the positive Grassmannian $Gr_{k,n}^{\geq 0}$ under the map ${Z}: Gr_{k,n}^{\geq 0} \to Gr_{k,k+m}$ induced by a positive linear map $Z:\mathbb{R}^n \to \mathbb{R}^{k+m}$. Motivated by a question of Hodges, Arkani-Hamed and Trnka introduced the amplituhedron in 2013 as a geometric object whose tilings conjecturally encode the BCFW recursion for computing scattering amplitudes. More specifically, the expectation was that one can compute scattering amplitudes in ${N}=4$ SYM by tiling the $m=4$ amplituhedron $A_{n,k,4}(Z)$ - that is, decomposing the amplituhedron into 'tiles' (closures of images of $4k$-dimensional cells of $Gr_{k,n}^{\geq 0}$ on which ${Z}$ is injective) - and summing the 'volumes' of the tiles. Also in 2013, Golden-Goncharov-Spradlin-Vergu-Volovich gave the first link between scattering amplitudes and cluster algebras, with Drummond-Foster-Gurdogan subsequently formulating the {cluster adjacency conjecture}. In this article we reveal and prove the deep mechanism behind `cluster phenomena' in tree-level scattering amplitudes. By connecting the BCFW recursion to a new cluster quasi-homomorphism on the Grassmannian $\Gr_{4,n}$, we prove the {cluster adjacency conjecture} for BCFW tiles, which says that each tile is a semialgebraic subset of the amplituhedron where a collection of compatible cluster variables take on definite signs. In particular, the facets of these tiles are cut out by compatible cluster variables. We also use the cluster description of BCFW tiles to prove the {BCFW tiling conjecture}, resolving the main original conjecture for the $m=4$ amplituhedron.

math.CO↗
arXiv2026

The m=2 amplituhedron and the hypersimplex: signs, clusters, triangulations, Eulerian numbers

The hypersimplex $Δ_{k+1,n}$ is the image of the positive Grassmannian $Gr^{\geq 0}_{k+1,n}$ under the moment map. It is a polytope of dimension $n-1$ in $\mathbb{R}^n$. Meanwhile, the amplituhedron ${A}_{n,k,2}(Z)$ is the projection of the positive Grassmannian $Gr^{\geq 0}_{k,n}$ into $Gr_{k,k+2}$ under a map $\tilde{Z}$ induced by a matrix $Z\in \text{Mat}_{n,k+2}^{>0}$. Introduced in the context of scattering amplitudes, it is not a polytope, and has dimension $2k$. Nevertheless, there seem to be remarkable connections between these two objects via T-duality, as was first noted by Lukowski--Parisi--Williams (LPW). In this paper we use ideas from oriented matroid theory, total positivity, and the geometry of the hypersimplex and positroid polytopes to obtain a deeper understanding of the amplituhedron. We show that the inequalities cutting out positroid polytopes -- images of positroid cells of $Gr^{\geq 0}_{k+1,n}$ under the moment map -- translate into sign conditions characterizing the T-dual Grasstopes -- images of positroid cells of $Gr^{\geq 0}_{k,n}$ under $\tilde{Z}$. Moreover, we subdivide the amplituhedron into chambers, just as the hypersimplex can be subdivided into simplices, with both chambers and simplices enumerated by the Eulerian numbers. We prove the main conjecture of (LPW): a collection of positroid polytopes is a triangulation of $Δ_{k+1, n}$ if and only if the collection of T-dual Grasstopes is a triangulation of ${A}_{n,k,2}(Z)$ for all $Z$. Moreover, we prove Arkani-Hamed--Thomas--Trnka's conjectural sign-flip characterization of ${A}_{n,k,2}(Z)$, and Lukowski--Parisi--Spradlin--Volovich's conjectures on $m=2$ cluster adjacency and on generalized triangles (images of $2k$-dimensional positroid cells which map injectively into ${A}_{n,k,2}(Z)$). Finally, we introduce new cluster structures in the amplituhedron.

math.CO↗
arXiv2026

Permutahedra, Lusztig varieties, degenerations, and subdivisions

We present an embedded (in $G/B$) degeneration of Lusztig varieties (which generalize type $A$ Hessenberg varieties) to certain unions of Richardson varieties, giving a simultaneous reproof (and extension) of results of Anderson--Tymoczko, Harada--Horiguchi--Masuda--Park, and Kim. Although torus-equivariant, the degeneration is not Gröbner. In the case that the Lusztig variety is the permutahedral toric variety, this degeneration provides a subdivision of the permutahedron into Bruhat interval polytopes, and we prove a more general result showing equivariant degenerations of projective toric varieties produce subdivisions of the moment polytope (as was shown in the Gröbner case by Sturmfels). A Gröbner degeneration would result in a {\em regular} subdivision, and despite our degeneration not being Gröbner we show in types $A,B,C$ that our subdivisions of the permutahedron are indeed regular.

math.AG↗
arXiv2026

Persistent Subdivisions of Coxeter Permutahedra

We investigate the realizations of Coxeter permutahedra which are also Coxeter matroid polytopes; these are polytopes of the form $\mathrm{conv}(W \cdot \mathbf{a})$ where $W$ is a finite Coxeter group acting on $\mathbb{R}^n$ and $\mathbf{a}$ is generic. Our main focus is how the geometric properties of $\mathrm{conv}(W \cdot \mathbf{a})$ change as $\mathbf{a}$ changes, with particular attention to persistent simplices, triangulations, and subdivisions.

math.CO↗
arXiv2026

Unexpected toric Richardson varieties

We prove that an open Richardson variety in the complete flag variety for $\mathrm{GL}_n$ is isomorphic to a torus if and only if the corresponding closed Richardson variety is toric. Such toric varieties can be classified in terms of the combinatorics of Bruhat intervals, and include many varieties of dimension larger than $n-1$. We give a combinatorial description of the corresponding polytopes, and compute several explicit examples.

math.AG↗
arXiv2026

Large induced acyclic and outerplanar subgraphs of 2-outerplanar graph

Albertson and Berman conjectured that every planar graph has an induced forest on half of its vertices. The best known lower bound, due to Borodin, is that every planar graph has an induced forest on two fifths of its vertices. In a related result, Chartran and Kronk, proved that the vertices of every planar graph can be partitioned into three sets, each of which induce a forest. We show tighter results for 2-outerplanar graphs. We show that every 2-outerplanar graph has an induced forest on at least half the vertices by showing that its vertices can be partitioned into two sets, each of which induces a forest. We also show that every 2-outerplanar graph has an induced outerplanar graph on at least two-thirds of its vertices, assuming that the connected components of the inner layer are two-connected.

math.CO↗
arXiv2026

Plabic Tangles and Cluster Promotion Maps

Inspired by the BCFW recurrence for tilings of the amplituhedron, we introduce the general framework of `plabic tangles' that utilizes plabic graphs to define rational maps between products of Grassmannians called `promotions'. The central conjecture of the paper is that promotion maps are quasi-cluster homomorphisms, which we prove for several classes of promotions. In order to define promotion maps, we utilize $m$-vector-relation configurations ($m$-VRCs) on plabic graphs. We relate $m$-VRCs to the degree (a.k.a `intersection number') of the amplituhedron map on positroid varieties and characterize all plabic trees with intersection number one and their VRCs. Finally, we show that promotion maps admit an operad structure and, supported by the class of `$4$-mass box' promotions, we point at new positivity properties for non-rational maps beyond cluster algebras. Promotion maps have important connections to the geometry and cluster structure of the amplituhedron and singularities of scattering amplitudes in planar $\mathcal{N}=4$ super Yang-Mills theory.

math.CO↗
arXiv2025

Grassmannian Geometries for Non-Planar On-Shell Diagrams

On-shell diagrams are gauge invariant quantities which play an important role in the description of scattering amplitudes. Based on the principles of generalized unitarity, they are given by products of elementary three-point amplitudes where the kinematics of internal on-shell legs are determined by cut conditions. In the ${\cal N}=4$ Super Yang-Mills (SYM) theory, the dual formulation for on-shell diagrams produces the same quantities as canonical forms on the Grassmannian $G(k,n)$. Most of the work in this direction has been devoted to the planar diagrams, which dominate in the large $N$ limit of gauge theories. On the mathematical side, planar on-shell diagrams correspond to cells of the positive Grassmannian $G_+(k,n)$ which have been very extensively studied in the literature in the past 20 years. In this paper, we focus on the non-planar on-shell diagrams which are relevant at finite $N$. In particular, we use the triplet formulation of Maximal-Helicity-Violating (MHV) on-shell diagrams to obtain certain regions in the Grassmannian $G(2,n)$. These regions are unions of positive Grassmannians with different orderings (referred to as oriented regions). We explore the features of these unions, and show that they are pseudo-positive geometries, in contrast to positive geometry of a single oriented region. For all non-planar diagrams which are \emph{internally planar} there always exists a strongly connected geometry, and for those that are \emph{irreducible}, there exists a geometry with no spurious facets. We also prove that the already known identity moves, square and sphere moves, form the complete set of identity moves for all MHV on-shell diagrams.

hep-th↗
arXiv2025

Permutahedron Triangulations via Total Linear Stability and the Dual Braid Group

For each finite Coxeter group $W$ and each standard Coxeter element of $W$, we construct a triangulation of the $W$-permutahedron. For particular realizations of the $W$-permutahedron, we show that this is a regular triangulation induced by a height function coming from the theory of total linear stability for Dynkin quivers. We also explore several notable combinatorial properties of these triangulations that relate the Bruhat order, the noncrossing partition lattice, and Cambrian congruences. Each triangulation gives an explicit mechanism for relating two different presentations of the corresponding braid group (the standard Artin presentation and Bessis's dual presentation). This is a step toward uniformly proving conjectural simple, explicit, and type-uniform presentations for the corresponding pure braid group.

math.CO↗
arXiv2025

Comparing cluster algebras on braid varieties

Braid varieties parametrize linear configurations of flags with transversality conditions dictated by positive braids. They include and generalize reduced double Bruhat cells, positroid varieties, open Bott-Samelson varieties, and Richardson varieties, among others. Recently, two cluster algebra structures were independently constructed in the coordinate rings of braid varieties: one using weaves and the other using Deodhar geometry. The main result of the article is that these two cluster algebras coincide. More generally, our comparative study matches the different concepts and results from each approach to the other, both on the combinatorial and algebraic geometric aspects.

math.AG↗
arXiv2025

BCFW tilings and cluster adjacency for the amplituhedron

In 2005, Britto, Cachazo, Feng and Witten gave a recurrence (now known as the BCFW recurrence) for computing scattering amplitudes in N=4 super Yang Mills theory. Arkani-Hamed and Trnka subsequently introduced the amplituhedron to give a geometric interpretation of the BCFW recurrence. Arkani-Hamed and Trnka conjectured that each way of iterating the BCFW recurrence gives a "triangulation" or "tiling" of the m=4 amplituhedron. In this article we prove the BCFW tiling conjecture of Arkani-Hamed and Trnka. We also prove the cluster adjacency conjecture for BCFW tiles of the amplituhedron, which says that facets of tiles are cut out by collections of compatible cluster variables for the Grassmannian Gr(4,n). Moreover we show that each BCFW tile is the subset of the Grassmannian where certain cluster variables have particular signs.

math.CO↗
arXiv2024

Dimer face polynomials in knot theory and cluster algebras

The set of perfect matchings of a connected bipartite plane graph $G$ has the structure of a distributive lattice, as shown by Propp, where the partial order is induced by the height of a matching. In this article, our focus is the dimer face polynomial of $G$, which is the height generating function of all perfect matchings of $G$. We connect the dimer face polynomial on the one hand to knot theory, and on the other to cluster algebras. We show that certain dimer face polynomials are multivariate generalizations of Alexander polynomials of links, highlighting another combinatorial view of the Alexander polynomial. We also show that an arbitrary dimer face polynomial is an $F$-polynomial in the cluster algebra whose initial quiver is dual to the graph $G$. As a result, we recover a recent representation theoretic result of Bazier-Matte and Schiffler that connects $F$-polynomials and Alexander polynomials, albeit from a very different, dimer-based perspective. As another application of our results, we also show that all nonvanishing Plücker coordinates on open positroid varieties are cluster monomials.

math.CO↗
arXiv2024

The Magic Number Conjecture for the $m=2$ amplituhedron and Parke-Taylor identities

The amplituhedron $A_{n,k,m}$ is a geometric object introduced in the context of scattering amplitudes in $N=4$ super Yang Mills. It generalizes the positive Grassmannian (when $n=k+m$), cyclic polytopes (when $k=1$), and the bounded complex of the cyclic hyperplane arrangement (when $m=1$). Of substantial interest are the tilings of the amplituhedron, which are analogous to triangulations of a polytope. Karp, Williams and Zhang (2020) observed that the known tilings of $A_{n,k,2}$ have cardinality ${n-2 \choose k}$ and the known tilings of $A_{n,k,4}$ have cardinality the Narayana number $\frac{1}{n-3}{n-3 \choose k+1}{n-3 \choose k}$; generalizing these observations, they conjectured that for even $m$ the tilings of $A_{n, k,m}$ have cardinality the MacMahon number, the number of plane partitions which fit inside a $k \times (n-k-m) \times \frac{m}{2}$ box. We refer to this prediction as the `Magic Number Conjecture'. In this paper we prove the Magic Number Conjecture for the $m=2$ amplituhedron: that is, we show that each tiling of $A_{n,k,2}$ has cardinality ${n-2 \choose k}$. We prove this by showing that all positroid tilings of the hypersimplex $Δ_{k+1,n}$ have cardinality ${n-2 \choose k}$, then applying T-duality. In addition, we give combinatorial necessary conditions for tiles to form a tiling of $A_{n,k,2}$; we give volume formulas for Parke-Taylor polytopes and certain positroid polytopes in terms of circular extensions of cyclic partial orders; and we prove new variants of the classical Parke-Taylor identities.

math.CO↗
arXiv2024

Braid variety cluster structures, I: 3D plabic graphs

We introduce $3$-dimensional generalizations of Postnikov's plabic graphs and use them to establish cluster structures for type $A$ braid varieties. Our results include known cluster structures on open positroid varieties and double Bruhat cells, and establish new cluster structures for type $A$ open Richardson varieties.

math.CO↗
arXiv2024

A cluster of results on amplituhedron tiles

The amplituhedron is a mathematical object which was introduced to provide a geometric origin of scattering amplitudes in $\mathcal{N}=4$ super Yang Mills theory. It generalizes \emph{cyclic polytopes} and the \emph{positive Grassmannian}, and has a very rich combinatorics with connections to cluster algebras. In this article we provide a series of results about tiles and tilings of the $m=4$ amplituhedron. Firstly, we provide a full characterization of facets of BCFW tiles in terms of cluster variables for $\mbox{Gr}_{4,n}$. Secondly, we exhibit a tiling of the $m=4$ amplituhedron which involves a tile which does not come from the BCFW recurrence -- the \emph{spurion} tile, which also satisfies all cluster properties. Finally, strengthening the connection with cluster algebras, we show that each standard BCFW tile is the positive part of a cluster variety, which allows us to compute the canonical form of each such tile explicitly in terms of cluster variables for $\mbox{Gr}_{4,n}$. This paper is a companion to our previous paper ``Cluster algebras and tilings for the $m=4$ amplituhedron''.

math.CO↗
arXiv2023

Demazure weaves for reduced plabic graphs (with a proof that Muller-Speyer twist is Donaldson-Thomas)

First, this article develops the theory of weaves and their cluster structures for the affine cones of positroid varieties. In particular, we explain how to construct a weave from a reduced plabic graph, show it is Demazure, compare their associated cluster structures, and prove that the conjugate surface of the graph is Hamiltonian isotopic to the Lagrangian filling associated to the weave. The T-duality map for plabic graphs has a surprising key role in the construction of these weaves. Second, we use the above established bridge between weaves and reduced plabic graphs to show that the Muller-Speyer twist map on positroid varieties is the Donaldson-Thomas transformation. This latter statement implies that the Muller-Speyer twist is a quasi-cluster automorphism. An additional corollary of our results is that target labeled seeds and the source labeled seeds are related by a quasi-cluster transformation.

math.CO↗