Root polytopes, Tutte polynomials, and a duality theorem for bipartite graphs

Kálmán, Tamás; Postnikov, Alexander

doi:10.1112/plms.12015

Cited by 30 publications

(58 citation statements)

References 12 publications

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“…(Here the transpose of the hypergraph H = (V, E) is the hypergraph H = (E, V ) with the roles of vertices and hyperedges interchanged.) In other words, I is an invariant of the bipartite graph Bip H. This fact is proven in [11] by noting that (essentially) the same polynomial may be obtained as the Ehrhart polynomial of the so called root polytope of Bip H. This depends, among other things, on the basic observation that spanning trees of a bipartite graph correspond to maximal simplices in its root polytope. We exploit the same connection to prove our main theorem.…”

Section: Introductionmentioning

confidence: 95%

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Hypergraph polynomials and the Bernardi process

Kálmán¹,

Tóthmérész²

2020

Algebraic Combinatorics

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Bernardi gave a formula for the Tutte polynomial T (x, y) of a graph, based on spanning trees and activities just like the original definition, but using a fixed ribbon structure to order the set of edges in a different way for each tree. The interior polynomial I is a generalization of T (x, 1) to hypergraphs. We supply a Bernardi-type description of I using a ribbon structure on the underlying bipartite graph G. Our formula works because it is determined by the Ehrhart polynomial of the root polytope of G in the same way as I is. To prove this we interpret the Bernardi process as a way of dissecting the root polytope into simplices, along with a shelling order. We also show that our generalized Bernardi process gives a common extension of bijections (and their inverses), constructed by Bernardi and further studied by Baker and Wang, between spanning trees and break divisors.

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Section: Introductionmentioning

confidence: 95%

“…Let H = (V, E) be a hypergraph so that Bip H is connected. For some fixed linear order on E we consider the generating function of internal inactivity, I H (ξ) = f ∈B E ξῑ (f ) , and call it the interior polynomial of H. By [9,Theorem 5.4] (see also [11,Subsection 2.2]), I H does not depend on the order.…”

Section: Algebraic Combinatorics Vol 3 #5 (2020)mentioning

confidence: 99%

Hypergraph polynomials and the Bernardi process

Kálmán¹,

Tóthmérész²

2020

Algebraic Combinatorics

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“…In the rest of the present paper, we discuss the γ -positivity and the real-rootedness of the h * -polynomial of B G when G is a bipartite graph. The edge polytope P G of a bipartite graph G is called the root polytope of G, and it is shown [25] that the h * -polynomial of P G coincides with the interior polynomial I G (x) of a hypergraph induced by G. First, we discuss interior polynomials introduced by Kálmán [24] and developed in many papers.…”

Section: Proposition 32 Letmentioning

confidence: 99%

“…In particular, edge polytopes give interesting examples in commutative algebra ( [18,[33][34][35]43]). Note that edge polytopes of bipartite graphs are called root polytopes and play important roles in the study of generalized permutohedra [41] and interior polynomials [25].…”

Section: Introductionmentioning

confidence: 99%

Reflexive polytopes arising from bipartite graphs with $$\gamma $$-positivity associated to interior polynomials

Ohsugi

Tsuchiya

2020

Sel. Math. New Ser.

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In this paper, we introduce polytopes $${\mathscr {B}}_G$$ B G arising from root systems $$B_n$$ B n and finite graphs G, and study their combinatorial and algebraic properties. In particular, it is shown that $${\mathscr {B}}_G$$ B G is reflexive if and only if G is bipartite. Moreover, in the case, $${\mathscr {B}}_G$$ B G has a regular unimodular triangulation. This implies that the $$h^*$$ h ∗ -polynomial of $${\mathscr {B}}_G$$ B G is palindromic and unimodal when G is bipartite. Furthermore, we discuss stronger properties, namely the $$\gamma $$ γ -positivity and the real-rootedness of the $$h^*$$ h ∗ -polynomials. In fact, if G is bipartite, then the $$h^*$$ h ∗ -polynomial of $${\mathscr {B}}_G$$ B G is $$\gamma $$ γ -positive and its $$\gamma $$ γ -polynomial is given by an interior polynomial (a version of the Tutte polynomial for a hypergraph). The $$h^*$$ h ∗ -polynomial is real-rooted if and only if the corresponding interior polynomial is real-rooted. From a counterexample to Neggers–Stanley conjecture, we construct a bipartite graph G whose $$h^*$$ h ∗ -polynomial is not real-rooted but $$\gamma $$ γ -positive, and coincides with the h-polynomial of a flag triangulation of a sphere.

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“…The Homfly polynomials of the three links (via the identity

Δ false(t false) = P false(1, t^{1 / 2} - t^{- 1 / 2} false)

, where

normalΔ

is the Alexander and

P

is the Homfly polynomial) induce partitions of this coefficient which do not coincide, but each can be derived from the appropriate set of hypertrees using the interior polynomial introduced by the first author in (see [, Corollary 1.2]. The proof also uses results from .)…”

Section: Introductionmentioning

confidence: 99%

Tight contact structures on Seifert surface complements

Kálmán

Mathews

2020

Journal of Topology

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We consider complements of standard Seifert surfaces of special alternating links. On these handlebodies, we use Honda's method to enumerate those tight contact structures whose dividing sets are isotopic to the link, and find their number to be the leading coefficient of the Alexander polynomial. The Euler classes of the contact structures are identified with hypertrees in a certain hypergraph. Using earlier work, this establishes a connection between contact topology and the Homfly polynomial. We also show that the contact invariants of our tight contact structures form a basis for sutured Floer homology. Finally, we relate our methods and results to Kauffman's formal knot theory.

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Root polytopes, Tutte polynomials, and a duality theorem for bipartite graphs

Cited by 30 publications

References 12 publications

Hypergraph polynomials and the Bernardi process

Hypergraph polynomials and the Bernardi process

Reflexive polytopes arising from bipartite graphs with $$\gamma $$-positivity associated to interior polynomials

Tight contact structures on Seifert surface complements

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