The HOMFLYPT skein algebra of the torus and the elliptic Hall algebra

Abstract

We give a generators and relations presentation of the HOMFLYPT skein algebra $H$ of the torus $T^2$, and we give an explicit description of the module corresponding to the solid torus. Using this presentation, we show that $H$ is isomorphic to the $t=q$ specialization of the elliptic Hall algebra of Burban and Schiffmann [BS12]. As an application, for an iterated cable $K$ of the unknot, we use the elliptic Hall algebra to construct a 3-variable polynomial that specializes to the $\lambda$-colored Homflypt polynomial of $K$. We show that this polynomial also specializes to one constructed by Cherednik and Danilenko [CD14] using the ${\mathfrak{gl}}_N$ double affine Hecke algebra. This proves one of the Connection Conjectures in [CD14].

arXiv commentv1: preliminary version, 36 pages, many figures. v2: minor edits and improvements to exposition

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Cited by in corpus (20)

• Toric braids and $(m,n)$-parking functions
• The Elliptic Hall algebra and the deformed Khovanov Heisenberg category
• Legendrian skein algebras and Hall algebras
• The Harish-Chandra isomorphism for quantum GL_2
• Exact structures and degeneration of Hall algebras
• Modules over plane curve singularities in any ranks and DAHA
• Iterated torus knots and double affine Hecke algebras
• Extremal weight projectors II
• Affinization of monoidal categories
• Quantum diagrammatics for $F_4$
• $(G{L}_k\times S_n)$-Modules of Multivariate Diagonal Harmonics
• KLR and Schur algebras for curves and semi-cuspidal representations
• Continuum Kac-Moody algebras
• Skein recursion for holomorphic curves and invariants of the unknot
• The Hall Algebras of Surfaces I
• Extremal weight projectors
• Elliptic Hall algebra on ${\mathbb{F}}_1$
• Power sum elements in the $G_2$ skein algebra
• Superpolynomials of algebraic links
• Dubrovnik Skein Theory and Power Sum Elements