The Framed Braid Group and 3-Manifolds

Ko, Ki Hyoung; Smolinsky, Lawrence

doi:10.2307/2159278

Cited by 17 publications

(34 citation statements)

References 5 publications

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“…The main purpose of the construction of Theorem 4.3, is to show that the natural invariants associated to the ribbon category coincide with the cocycle invariants defined in Section 3. It is in fact possible to bypass this construction and prove that the twisting morphsism θ α and the braiding morphism c α 2,2 induce a representation of the infinite framed braid group (the inductive limit of the framed braid groups F B n of [18]) similarly to [26], using the linear map Φ b of Section 5 (given below). It is convenient, though, to define a ribbon category from a ternary self-distributive structure and a ternary 2-cocycle since this is suitable for a generalization to multiple compatible self-distributive structures.…”

Section: Ribbon Categories From Self-distributive Ternary Operationsmentioning

confidence: 99%

“…Let (X, T ) be a ternary self-distributive object arising from a set-theoretic ternary quandle Q as above, and [α] ∈ H 2 (Q, A) be fixed. A framed link is represented by the closure of an element b ∈ F B n of the framed braid group on n ribbons [18] where, since twisting of the ribbon and crossings commute, it is assumed that the twists are on top of the braid. Using the same notation of [18],…”

Section: The Ribbon Cocycle Invariant Is a Quantum Invariantmentioning

confidence: 99%

“…A framed link is represented by the closure of an element b ∈ F B n of the framed braid group on n ribbons [18] where, since twisting of the ribbon and crossings commute, it is assumed that the twists are on top of the braid. Using the same notation of [18],…”

Section: The Ribbon Cocycle Invariant Is a Quantum Invariantmentioning

confidence: 99%

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Quantum invariants of framed links from ternary self-distributive cohomology

Zappala

2021

Preprint

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The ribbon cocycle invariant is defined by means of a partition function using ternary cohomology of self-distributive structures (TSD) and colorings of ribbon diagrams of a framed link, following the same paradigm introduced by Carter, Jelsovsky, Kamada, Langfor and Saito in Transactions of the American Mathematical Society 2003;355(10):3947-89, for the quandle cocycle invariant. In this article we show that the ribbon cocycle invariant is a quantum invariant. We do so by constructing a ribbon category from a TSD set whose twisting and braiding morphisms entail a given TSD 2-cocycle. Then we show that the quantum invariant naturally associated to this braided category coincides with the cocycle invariant. We generalize this construction to symmetric monoidal categories and provide classes of examples obtained from Hopf monoids and Lie algebras. We further introduce examples from Hopf-Frobenius algebras, objects studied in quantum computing. EMANUELE ZAPPALA 7.5. Construction of ribbon categories 36 8. Invariants of framed links in symmetric monoidal categories 43 Appendix A. Examples from ternary augmented racks 45 References 48

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Section: Ribbon Categories From Self-distributive Ternary Operationsmentioning

confidence: 99%

Section: The Ribbon Cocycle Invariant Is a Quantum Invariantmentioning

confidence: 99%

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Quantum invariants of framed links from ternary self-distributive cohomology

Zappala

2021

Preprint

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“…The above relations may be familiar to many readers. They appear in the study of framed, or ribbon, braid groups, introduced in [14]. A more in-depth and mathematically rigourous treatment of framed braids or ribbon braids can be found in [15,16], among others.…”

Section: Thementioning

confidence: 99%

“…necessary. In contrast, for framed braids in an infinitely long strip as in [14] more complicated crossing conditions are needed.…”

Section: These Conventions Givementioning

confidence: 99%

Graphical Calculus for the Double Affine Q-Dependent Braid Group

Burella

Watts

Pasquier

et al. 2013

Ann. Henri Poincaré

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We define a double affine $Q$-dependent braid group. This group is constructed by appending to the braid group a set of operators $Q_i$, before extending it to an affine $Q$-dependent braid group. We show specifically that the elliptic braid group and the double affine Hecke algebra (DAHA) can be obtained as quotient groups. Complementing this we present a pictorial representation of the double affine $Q$-dependent braid group based on ribbons living in a toroid. We show that in this pictorial representation we can fully describe any DAHA. Specifically, we graphically describe the parameter $q$ upon which this algebra is dependent and show that in this particular representation $q$ corresponds to a twist in the ribbon

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An isomorphism theorem for Yokonuma–Hecke algebras and applications to link invariants

Jacon

d'Andecy

2015

Math. Z.

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We develop several applications of the fact that the Yokonuma-Hecke algebra of the general linear group GL is isomorphic to a direct sum of matrix algebras associated to Iwahori-Hecke algebras of type A. This includes a description of the semisimple and modular representation theory of the Yokonuma-Hecke algebras of GL and a complete classification of all the Markov traces for them. Finally, from these Markov traces, we construct 3-variables polynomials which are invariants for framed and classical knots and links, and investigate their properties with the help of the isomorphism.

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The Framed Braid Group and 3-Manifolds

Cited by 17 publications

References 5 publications

Quantum invariants of framed links from ternary self-distributive cohomology

Quantum invariants of framed links from ternary self-distributive cohomology

Graphical Calculus for the Double Affine Q-Dependent Braid Group

An isomorphism theorem for Yokonuma–Hecke algebras and applications to link invariants

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