HOMFLYPT homology for links in handlebodies via type A Soergel bimodules

Rose, David E. V.; Tubbenhauer, Daniel

doi:10.4171/qt/152

Cited by 6 publications

(4 citation statements)

References 28 publications

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“…Remark 4.34 Our proof of Lemma 4.33 is directly inspired by [RT19]: As pointed out in that paper, the handlebody closing (3-4) can be interpreted, in the appropriate algebraic framework, by putting an idempotent on bottom and top. Moreover and alternatively to the usage of (growing) symmetric powers, one might want to associate Verma modules to the core strands as in the g = 1 case, see [ILZ21].…”

Section: Definition 432 We Call Parametersmentioning

confidence: 99%

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Handlebody diagram algebras

Tubbenhauer,

Vaz

2021

Preprint

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In this paper we study handlebody versions of classical diagram algebras, most prominently, handlebody versions of Temperley-Lieb, blob, Brauer/BMW, Hecke and Ariki-Koike algebras. Moreover, motivated by Green-Kazhdan-Lusztig's theory of cells, we reformulate the notion of (sandwich, inflated or affine) cellular algebras. We explain this reformulation and how all of the above algebras are part of this theory. Contents1. Introduction 1 2. A generalization of cellularity 4 3. Handlebody braid and Coxeter groups 11 4. Handlebody Temperley-Lieb and blob algebras 16 5. Handlebody Brauer and BMW algebras 28 6. Handlebody Hecke and Ariki-Koike algebras 36 References 41

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Section: Definition 432 We Call Parametersmentioning

confidence: 99%

“…For example, for g = 1 [GL97] and [OR07] construct link invariants from Markov traces, and these link invariants admit categorifications [WW11]. For g > 1 [RT19] takes a few first steps towards categorical handlebody link invariants, but this direction appears to be widely open otherwise.…”

Section: (B)mentioning

confidence: 99%

Handlebody diagram algebras

Tubbenhauer,

Vaz

2021

Preprint

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“…(b) Quantum Verma Howe duality originated in our attempt to understand invariants of links in handlebodies coming from handlebody braid groups, see e.g. [HOL02], [Ver98], [RT21], [TV21]. The associated Howe duality should involve a tensor product of Verma modules for the cores of the handlebodies, and also a tensor product of finite dimensional modules for the strands of the braids.…”

Section: Introductionmentioning

confidence: 99%

Verma Howe duality and LKB representations

Lacabanne¹,

Tubbenhauer²,

Vaz³

2022

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We establish a version of quantum Howe duality with two general linear quantum enveloping algebras that involves a tensor product of Verma modules. We prove that the (colored higher) LKB representations arise from this duality and use this description to show that they are simple as modules for various subgroups of the braid group, including the pure braid group.1A. Schur-Weyl(-Brauer) and Howe dualities. Three main themes in Weyl's seminal book "The classical groups" [Wey97] are the study of polynomial invariants for actions of the eponymous classical groups, and, more or less equivalent, decomposition of the tensor algebra for such an action, and, again more or less equivalent, the description of the invariants in the tensor algebra.The two most prominent examples that fit into Weyl's setting are the celebrated Schur-Weyl duality [Sch01] for tensor invariants of GL m (C) and Brauer duality [Bra37] for tensor invariants of O m (C) and SP m (C) (for the symplectic group m is even). Both of these were studied by using commuting actions of GL m (C), and O m (C), SP m (C) on one side and the symmetric group S n and the Brauer algebra, respectively, on the other side, both acting on a tensor product of the defining representation of the classical groups. In this commuting-action-approach, Schur-Weyl duality for example essentially reads:(A) There are commuting actions of GL m (C) and S n on (C m ) ⊗n .(B) The two actions generate each others centralizer.(C) The GL m (C)-S n bimodule (C m ) ⊗n can be explicitly decomposed into a direct sum of simple GL m (C) modules tensored with simple S n modules. A statement of this form is what we call a double centralizer (a.k.a. double commutant) approach.Howe [How89], [How95] studied polynomial invariants, e.g. via symmetric powers, of classical groups using a double centralizer approach, and the resulting dualities are called Howe dualities

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“…This is already a reason to develop efficient calculus for associated knot polynomials.Apart from that, the Khovanov polynomial of a satellite is actually the polynomial of a knot in a solid torus. Thus we get another approach to the knots in the simplest handled body, for which these invariants invariants are yet little studied [4].…”

Section: Introductionmentioning

confidence: 99%

Khovanov polynomials for satellites and asymptotic adjoint polynomials

Anokhina,

Morozov,

Popolitov

2021

Preprint

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We compute explicitly the Khovanov polynomials (using the computer program from katlas.org) for the two simplest families of the satellite knots, which are the twisted Whitehead doubles and the twostrand cables. We find that a quantum group decomposition for the HOMFLY polynomial of a satellite knot can be extended to the Khovanov polynomial, whose quantum group properties are not manifest. Namely, the Khovanov polynomial of a twisted Whitehead double or two-strand cable (the two simplest satellite families) can be presented as a naively deformed linear combination of the pattern and companion invariants. For a given companion, the satellite polynomial "smoothly" depends on the pattern but for the "jump" at one critical point defined by the s-invariant of the companion knot. A similar phenomenon is known for the knot Floer homology and τ -invariant for the same kind of satellites

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HOMFLYPT homology for links in handlebodies via type A Soergel bimodules

Cited by 6 publications

References 28 publications

Handlebody diagram algebras

Handlebody diagram algebras

Verma Howe duality and LKB representations

Khovanov polynomials for satellites and asymptotic adjoint polynomials

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