Knot theory and cluster algebras

Bazier-Matte, Véronique; Schiffler, Ralf

doi:10.1016/j.aim.2022.108609

Cited by 8 publications

(24 citation statements)

References 42 publications

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“…Remark We recover and generalize the main result in [5], where the authors obtain the similar result for Dynkin quivers (without potentials). In such cases, the Newton polytope is the so‐called generalized associahedron [23].…”

Section: Examplessupporting

confidence: 85%

Tropical F$F$‐polynomials and general presentations

Fei

2023

Journal of London Math Soc

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We introduce the tropical F$F$‐polynomial fM$f_M$ of a quiver representation M$M$. We study its interplay with the general presentation for any finite‐dimensional basic algebra. We give an interpretation of evaluating fM$f_M$ at a weight vector. As a consequence, we give a presentation of the Newton polytope sans-serifNfalse(Mfalse)${\sf N}(M)$ of M$M$. We study the dual fan and 1‐skeleton of sans-serifNfalse(Mfalse)${\sf N}(M)$. We propose an algorithm to determine the generic Newton polytopes, and show that it works for path algebras. As an application, we give a representation‐theoretic interpretation of Fock–Goncharov's duality pairing. We give an explicit construction of dual clusters, which consists of real Schur representations. We specialize the above general results to the cluster‐finite algebras and the preprojective algebras of Dynkin type.

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Section: Examplessupporting

confidence: 85%

Tropical F$F$‐polynomials and general presentations

Fei

2023

Journal of London Math Soc

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“…Geodesic functions enjoy the classical skein relations and semiclassical Poisson relations (Goldman brackets); using these two relations we can construct any other geodesic function for a smooth Riemann surface of genus g > 1 out of 2g + 1 specially chosen geodesics. 1 Say, having n − 1 geodesic functions G i,i+1 , i = 1, . .…”

Section: Figure 12mentioning

confidence: 99%

“…Note that these loops form a chain with intersection numbers one for two consecutive loops and zero otherwise. We construct a matrix U ∈ A 8 with U i,i+1 , i = [1,7] given by the corresponding element of sequence (8.31) and all other terms computed using the skein and Poisson relations…”

Section: Figure 12mentioning

confidence: 99%

Log-Canonical Coordinates for Symplectic Groupoid and Cluster Algebras

Chekhov

Shapiro

2022

International Mathematics Research Notices

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Using Fock–Goncharov higher Teichmüller space variables we derive log-canonical coordinate representation for entries of general symplectic leaves of the $\mathcal A_n$ groupoid of upper-triangular matrices and, in a more general setting, of higher-dimensional symplectic leaves for algebras governed by the reflection equation with the trigonometric $R$-matrix. The obtained results are in a perfect agreement with the previously obtained Poisson and quantum representations of groupoid variables for $\mathcal A_3$ and $\mathcal A_4$ in terms of geodesic functions for Riemann surfaces with holes. We realize braid-group transformations for $\mathcal A_n$ via sequences of cluster mutations in the special $\mathcal A_n$-quiver. We prove the groupoid relations for normalized quantum transport matrices and, as a byproduct, obtain the Goldman bracket in the semiclassical limit. We prove the quantum algebraic relations of transport matrices for arbitrary (cyclic or acyclic) directed planar network.Dedicated to the memory of a great mathematician and person, Boris Dubrovin.

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“…This construction provides a large degree of freedom in the choice of the parameters defining these affine subspaces, and actually produces all polytopes whose normal fan is affinely equivalent to that of Loday's associahedron [60] (see Subsection 2.1). These realizations were then extended by Bazier‐Matte, Chapelier‐Laget, Douville, Mousavand, Thomas, and Yıldırım [17] in the context of finite‐type cluster algebras using tools from representation theory of quivers. More precisely, they fix a finite‐type cluster algebra

{\mathcal {A}}

and consider the real euclidean space

\mathbb {R}^{\mathcal {V}}

indexed by the set

{\mathcal {V}}

of cluster variables of

{\mathcal {A}}

.…”

Section: Introductionmentioning

confidence: 99%

Associahedra for finite‐type cluster algebras and minimal relations between g‐vectors

Padrol,

Palu,

Pilaud

et al. 2023

Proceedings of London Math Soc

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We show that the mesh mutations are the minimal relations among the ‐vectors with respect to any initial seed in any finite‐type cluster algebra. We then use this algebraic result to derive geometric properties of the ‐vector fan: we show that the space of all its polytopal realizations is a simplicial cone, and we then observe that this property implies that all its realizations can be described as the intersection of a high‐dimensional positive orthant with well‐chosen affine spaces. This sheds a new light on and extends earlier results of Arkani‐Hamed, Bai, He, and Yan in type and of Bazier‐Matte, Chapelier‐Laget, Douville, Mousavand, Thomas, and Yıldırım for acyclic initial seeds. Moreover, we use a similar approach to study the space of polytopal realizations of the ‐vector fans of another generalization of the associahedron: nonkissing complexes (also known as support ‐tilting complexes) of gentle algebras. We show that the space of realizations of the nonkissing fan is simplicial when the gentle bound quiver is brick and 2‐acyclic, and we describe in this case its facet‐defining inequalities in terms of mesh mutations. Along the way, we prove algebraic results on 2‐Calabi–Yau triangulated categories, and on extriangulated categories that are of independent interest. In particular, we prove, in those two setups, an analogue of a result of Auslander on minimal relations for Grothendieck groups of module categories.

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Knot theory and cluster algebras

Cited by 8 publications

References 42 publications

Tropical F$F$‐polynomials and general presentations

Tropical F$F$‐polynomials and general presentations

Log-Canonical Coordinates for Symplectic Groupoid and Cluster Algebras

Associahedra for finite‐type cluster algebras and minimal relations between g‐vectors

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