Alexander invariants of ribbon tangles and planar algebras

Damiani, Celeste; Florens, Vincent

doi:10.2969/jmsj/75267526

Cited by 9 publications

(7 citation statements)

References 15 publications

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“…This generalizes a phenomenon that has been observed in the special case of homology cobordisms [FM15]: see Remark 6.4. We conclude by mentioning that the proof of this formula can be adapted to the situation of tangles, which implies a similar relationship between the Alexander representation of tangles constructed in [BCF15] (see also [Arc10], [DF16]) and the functor of Cimasoni and Turaev.…”

Section: Introductionmentioning

confidence: 58%

A functorial extension of the Magnus representation to the category of three-dimensional cobordisms

Florens

Massuyeau

Rodrigo³

2018

Fund. Math.

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Let R be an integral domain and G be a subgroup of its group of units. We consider the category Cob G of 3-dimensional cobordisms between oriented surfaces with connected boundary, equipped with a representation of their fundamental group in G. Under some mild conditions on R, we construct a monoidal functor from Cob G to the category pLagr R consisting of "pointed Lagrangian relations" between skew-Hermitian R-modules. We call it the "Magnus functor" since it contains the Magnus representation of mapping class groups as a special case. Our construction is inspired from the work of Cimasoni and Turaev on the extension of the Burau representation of braid groups to the category of tangles. It can also be regarded as a G-equivariant version of a TQFT-like functor that has been described by Donaldson. The study and computation of the Magnus functor is carried out using classical techniques of low-dimensional topology. When G is a free abelian group and R = Z[G] is the group ring of G, we relate the Magnus functor to the "Alexander functor" (which has been introduced in a prior work using Alexander-type invariants), and we deduce a factorization formula for the latter.

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Section: Introductionmentioning

confidence: 58%

A functorial extension of the Magnus representation to the category of three-dimensional cobordisms

Florens

Massuyeau

Rodrigo³

2018

Fund. Math.

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“…Inspired by this and the Burau representation (and cf. [DF18] and references therein) we proceed as follows.…”

Section: On Local Representationsmentioning

confidence: 99%

Generalisations of Hecke algebras from Loop Braid Groups

Damiani,

Martin,

Rowell

2020

Preprint

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We introduce a generalisation LHn of the ordinary Hecke algebras informed by the loop braid group LBn and the extension of the Burau representation thereto. The ordinary Hecke algebra has many remarkable arithmetic and representation theoretic properties, and many applications. We show that LHn has analogues of several of these properties. In particular we introduce a class of local (tensor space/functor) representations of the braid group derived from a meld of the (nonfunctor) Burau representation [Bur35] and the (functor) Rittenberg representations [MR92], here thus called Burau-Rittenberg representations. In its most supersymmetric case somewhat mystical cancellations of anomalies occur so that the Burau-Rittenberg representation extends to a loop Burau-Rittenberg representation. And this factors through LHn. Let SPn denote the corresponding (not necessarily proper) quotient algebra, k the ground ring, and t ∈ k the loop-Hecke parameter. We prove the following:(1) LHn is finite dimensional over a field.(2) The natural inclusion LBn ֒→ LBn+1 passes to an inclusion SPn ֒→ SPn+1.(3) Over k = C, SPn/rad is generically the sum of simple matrix algebras of dimension (and Bratteli diagram) given by Pascal's triangle. (Specifically SPn/rad ∼ = CSn/e 1 (2,2) where Sn is the symmetric group and e 1 (2,2) is a λ = (2, 2) primitive idempotent.) (4) We determine the other fundamental invariants of SPn representation theory: the Cartan decomposition matrix; and the quiver, which is of type-A. (5) the structure of SPn is independent of the parameter t, except for t = 1. (6) For t 2 = 1 then LHn ∼ = SPn at least up to rank n = 7 (for t = −1 they are not isomorphic for n > 2; for t = 1 they are not isomorphic for n > 1). Finally we discuss a number of other intriguing points arising from this construction in topology, representation theory and combinatorics.

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“…In the case where a tangle has only one component, we recover the Alexander polynomial. One can obtain a topological interpretation of Γ-calculus along the lines of the arguments in [CT05] and [DF16] but we will not pursue it here. On a computer, Γ-calculus is quite simple to implement (see the Appendix) and it also runs faster than current algorithms that compute the Alexander polynomial.…”

Section: Introductionmentioning

confidence: 99%

On Meta-monoids and the Fox-Milnor Condition

Vo¹

2017

Preprint

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In this paper we introduce an algebraic structure known as meta-monoids which is suited for the study of knot theory. We define a particular meta-monoid called Γ-calculus that gives an Alexander invariant of tangles. Using the language of Γ-calculus, we rederive certain important properties of the Alexander polynomial, most notably the Fox-Milnor condition on the Alexander polynomials of ribbon knots [Lic97, FM66]. We argue that our proof has some potential for generalization which may help tackle the slice-ribbon conjecture. In a sense this paper is an extension of [BNS13].

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Alexander invariants of ribbon tangles and planar algebras

Cited by 9 publications

References 15 publications

A functorial extension of the Magnus representation to the category of three-dimensional cobordisms

A functorial extension of the Magnus representation to the category of three-dimensional cobordisms

Generalisations of Hecke algebras from Loop Braid Groups

On Meta-monoids and the Fox-Milnor Condition

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