Positive basis for surface skein algebras

Thurston, Dylan P.

doi:10.1073/pnas.1313070111

Cited by 65 publications

(39 citation statements)

References 31 publications

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“…1.1] in general. We further expect agreement with a quantum version of the bracelets basis of [MSW13] -this would prove [Thu14,Conj. 4.20].…”

Section: Introductionsupporting

confidence: 56%

Positivity for quantum cluster algebras

Davison

2018

Ann. of Math. (2)

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Abstract. Building on work by Kontsevich, Soibelman, Nagao and Efimov, we prove the positivity of quantum cluster coefficients for all skew-symmetric quantum cluster algebras, via a proof of a conjecture first suggested by Kontsevich on the purity of mixed Hodge structures arising in the theory of cluster mutation of spherical collections in 3-Calabi-Yau categories. The result implies positivity, as well as the stronger Lefschetz property conjectured by Efimov, and also the classical positivity conjecture of Fomin and Zelevinsky, recently proved by Lee and Schiffler. Closely related to these results is a categorified "no exotics" type theorem for cohomological Donaldson-Thomas invariants, which we discuss and prove in the appendix.

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“…1.1] in general. We further expect agreement with a quantum version of the bracelets basis of [MSW13] -this would prove [Thu14,Conj. 4.20].…”

Section: Introductionsupporting

confidence: 56%

Positivity for quantum cluster algebras

Davison

2018

Ann. of Math. (2)

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“…The Laurent polynomials associated to bangles and bracelets can be computed via Definition 5.19. One can easily check that the obtained formulae coincide with ones for surfaces (see [MSW2,T2]): if γ is a semi-closed curve, then x Bang k (γ) = (x γ ) k , x Brac k (γ) = T k (x γ ) and x Band k (γ) = U k (x γ ), where T k is the Chebyshev polynomial of the first kind defined by initial conditions T 0 (x) = 2, T 1 (x) = x and recurrent relation T k (x) = xT k−1 (x) − T k−2 (x), and U k is the Chebyshev polynomial of the second kind defined by initial conditions U 0 (x) = 1, U 1 (x) = x and recurrent relation U k (x) = xU k−1 (x) − U k−2 (x).…”

Section: 5mentioning

confidence: 61%

Bases for cluster algebras from orbifolds

Felikson

Tumarkin

2017

Advances in Mathematics

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Abstract. We generalize the construction of the bracelet and bangle bases defined in [MSW2] and the band basis defined in [T2] to cluster algebras arising from orbifolds. We prove that the bracelet bases are positive, and the bracelet basis for the affine cluster algebra of type C (1) n is atomic. We also show that cluster monomial bases of all skewsymmetrizable cluster algebras of finite type are atomic.

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“…For generalizations to the skew-symmetrizable case see [7,8]. The closely related surface skein algebras were studied in [21,30].…”

Section: Introductionmentioning

confidence: 99%

Snake graph calculus and cluster algebras from surfaces II: self-crossing snake graphs

Çanakçı

Schiffler

2015

Math. Z.

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Snake graphs appear naturally in the theory of cluster algebras. For cluster algebras from surfaces, each cluster variable is given by a formula whose terms are parametrized by the perfect matchings of a snake graph. In this paper, we continue our study of snake graphs from a combinatorial point of view. We advance the study of abstract snake graphs by introducing the notions of abstract band graphs, self-crossings of abstract snake graphs as well as their resolutions. We show that there is a bijection between the set of perfect matchings of crossing or self-crossing snake graphs and the set of perfect matchings of the resolution of the crossing. In the situation where the snake and band graphs are coming from arcs and loops in a surface without punctures, we obtain a new proof of skein relations in the corresponding cluster algebra.

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Positive basis for surface skein algebras

Abstract: We show that the twisted SL 2 skein algebra of a surface has a natural basis (the bracelets basis) that is positive, in the sense that the structure constants for multiplication are positive integers.character variety | canonical basis | Jones polynomial | cluster algebras

Cited by 65 publications

References 31 publications

Positivity for quantum cluster algebras

Positivity for quantum cluster algebras

Bases for cluster algebras from orbifolds

Snake graph calculus and cluster algebras from surfaces II: self-crossing snake graphs

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