Cluster structure on genus 2 spherical DAHA: seven-colored flower

Semeon Arthamonov; Leonid Chekhov; Philippe Di Francesco; Rinat Kedem; Gus Schrader; Alexander Shapiro; Michael Shapiro

Cluster structure on genus 2 spherical DAHA: seven-colored flower

Semeon Arthamonov, Leonid Chekhov, Philippe Di Francesco, Rinat Kedem, Gus Schrader, Alexander Shapiro, Michael Shapiro

School of Mathematics

Research output: Working paper › Preprint

Abstract

We construct an embedding of the Arthamonov–Shakirov algebra of genus 2 knot operators into the quantized coordinate ring of the cluster Poisson variety of exceptional finite mutation type X₇. The embedding is equivariant with respect to the action of the mapping class group of the closed surface of genus 2. The cluster realization of the mapping class group action leads to a formula for the coefficient of each monomial in the genus 2 Macdonald polynomial of type A₁ as sum over lattice points in a convex polyhedron in 7-dimensional space.

Original language	English
Publisher	ArXiv
Number of pages	27
Publication status	Published - 25 Feb 2024

Access to Document

https://arxiv.org/abs/2402.16074

Cite this

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title = "Cluster structure on genus 2 spherical DAHA: seven-colored flower",

abstract = "We construct an embedding of the Arthamonov–Shakirov algebra of genus 2 knot operators into the quantized coordinate ring of the cluster Poisson variety of exceptional finite mutation type X7. The embedding is equivariant with respect to the action of the mapping class group of the closed surface of genus 2. The cluster realization of the mapping class group action leads to a formula for the coefficient of each monomial in the genus 2 Macdonald polynomial of type A1 as sum over lattice points in a convex polyhedron in 7-dimensional space.",

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AU - Chekhov, Leonid

AU - Di Francesco, Philippe

AU - Kedem, Rinat

AU - Schrader, Gus

AU - Shapiro, Alexander

AU - Shapiro, Michael

PY - 2024/2/25

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N2 - We construct an embedding of the Arthamonov–Shakirov algebra of genus 2 knot operators into the quantized coordinate ring of the cluster Poisson variety of exceptional finite mutation type X7. The embedding is equivariant with respect to the action of the mapping class group of the closed surface of genus 2. The cluster realization of the mapping class group action leads to a formula for the coefficient of each monomial in the genus 2 Macdonald polynomial of type A1 as sum over lattice points in a convex polyhedron in 7-dimensional space.

AB - We construct an embedding of the Arthamonov–Shakirov algebra of genus 2 knot operators into the quantized coordinate ring of the cluster Poisson variety of exceptional finite mutation type X7. The embedding is equivariant with respect to the action of the mapping class group of the closed surface of genus 2. The cluster realization of the mapping class group action leads to a formula for the coefficient of each monomial in the genus 2 Macdonald polynomial of type A1 as sum over lattice points in a convex polyhedron in 7-dimensional space.

M3 - Preprint

BT - Cluster structure on genus 2 spherical DAHA: seven-colored flower

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