F-polynomial formula from continued fractions

Rabideau, Michelle

doi:10.1016/j.jalgebra.2018.05.019

Cited by 9 publications

(10 citation statements)

References 11 publications

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“…There are a few quantities defined on continued fractions which we will use often in this paper. These can be found in [LS19], [CS18], and [Rab18] (among others) defined only for positive or even continued fractions. We extend them here to all continued fractions.…”

Section: Proposition 21 ([Hw79][ls19]mentioning

confidence: 99%

Cluster Algebras and the HOMFLY Polynomial

Yacavone

2019

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Recently, it has been shown that the Jones polynomial, in [LS19], and the Alexander polynomial, in [NT18], of rational knots can be obtained by specializing F -polynomials of cluster variables. At the core of both results are continued fractions, which parameterize rational knots and are used to obtain cluster variables, by way of snake graphs in the case of [LS19], or ancestral triangles in the case of [NT18]. In this paper, we use path posets, another structure parameterized by continued fractions, to directly generalize [LS19]'s construction to a specialization yielding the HOMFLY polynomial, which generalizes both the Jones and Alexander polynomials.Code exploring the results of this paper can be found at: https://github.com/m-yac/F-polys.

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Section: Proposition 21 ([Hw79][ls19]mentioning

confidence: 99%

Cluster Algebras and the HOMFLY Polynomial

Yacavone

2019

Preprint

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“…Remark 2.1.1. The connection between these posets and F -polynomials in cluster algebras has been noted in [MSW11], [Rab18], [BG21], and [Cla20]. These posets were called "fence posets" in [MSS21] and "piece-wise linear posets" in [BG21].…”

Section: Combinatorial Interpretationsmentioning

confidence: 99%

$q$-Rationals and Finite Schubert Varieties

Ovenhouse¹

2021

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The classical q-analogue of the integers was recently generalized by Morier-Genoud and Ovsienko to give qanalogues of rational numbers. Some combinatorial interpretations are already known, namely as the rank generating functions for certain partially ordered sets. We review some of these interpretations, and additionally give a slightly novel approach in terms of planar graphs called snake graphs. Using the snake graph approach, we show that the numerators of q-rationals count the sizes of certain varieties over finite fields, which are unions of open Schubert cells in some Grassmannian.

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“…, G k ) is the single edge e j if j > k. The decomposition is in fact obtained by deleting the sign-changed tiles. Following [33], once we choose the minimal perfect matching P − of a snake graph G, then the minimal matching P − | H i of a subsnake graph H i is either the matching which inherits from P − or the union of the matching which inherits from P − and a unique interior edge.…”

Section: Preliminarymentioning

confidence: 99%

“…For simplicity, let j ℓ=i x ℓ = 1 for j < i. The following results from [33] will play an important role in the subsequent proof. In [33] Rabideau required that S G ∈ P − for a labeled snake graph G. Lemma 2.8.…”

Section: Preliminarymentioning

confidence: 99%

“…In particular, identities in the cluster algebra have been expressed in terms of snake graphs. Rabideau used continued fractions to give a combinatorial formula for F -polynomials [33] in cluster algebras with principal coefficients from surfaces. Rabideau and Schiffler proved the constant numerator conjecture for Markov numbers via snake graphs and continued fractions [34].…”

Section: Introductionmentioning

confidence: 99%

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Hernandez-Leclerc modules and snake graphs

Duan,

Li,

Luo

2020

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In 2010, Hernandez and Leclerc studied connections between representations of quantum affine algebras and cluster algebras. In 2019, Brito and Chari defined a family of modules over quantum affine algebras, called Hernandez-Leclerc modules. We characterize the highest ℓ-weight monomials of Hernandez-Leclerc modules. We give a non-recursive formula of q-characters of Hernandez-Leclerc modules using snake graphs, which involves an explicit formula for F -polynomials. We also give a new recursive formula of q-characters of Hernandez-Leclerc modules.

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F-polynomial formula from continued fractions

Cited by 9 publications

References 11 publications

Cluster Algebras and the HOMFLY Polynomial

Cluster Algebras and the HOMFLY Polynomial

$q$-Rationals and Finite Schubert Varieties

Hernandez-Leclerc modules and snake graphs

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