Categorification of the Colored Jones Polynomial and Rasmussen Invariant of Links

Beliakova, Anna; Wehrli, Stephan M.

doi:10.4153/cjm-2008-053-1

Cited by 68 publications

(121 citation statements)

References 9 publications

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“…It is important to note that our definition disagrees with some previous categorifications based on different axiomatizations of the Jones-Wenzl projectors such as the dimension axiom [17] (for related work see [5] and [11]):…”

Section: Universal Projectors and Statement Of The Main Theoremcontrasting

confidence: 58%

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Categorification of the Jones–Wenzl projectors

Cooper¹,

Krushkal²

2012

Quantum Topol.

148

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Abstract. The Jones-Wenzl projectors p n play a central role in quantum topology, underlying the construction of SU.2/ topological quantum field theories and quantum spin networks. We construct chain complexes P n , whose graded Euler characteristic is the "classical" projector p n in the Temperley-Lieb algebra. We show that the P n are idempotents and uniquely defined up to homotopy. Our results fit within the general framework of Khovanov's categorification of the Jones polynomial. Consequences of our construction include families of knot invariants corresponding to higher representations of U q sl.2/ and a categorification of quantum spin networks. We introduce 6j -symbols in this context.

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Section: Universal Projectors and Statement Of The Main Theoremcontrasting

confidence: 58%

“…Note that the included computations imply that our work is different from the previously defined categorification of the colored Jones polynomial [17] (see also [5]). See Section 4.1 for further discussion.…”

Section: Algebra Categorymentioning

confidence: 90%

Categorification of the Jones–Wenzl projectors

Cooper¹,

Krushkal²

2012

Quantum Topol.

148

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“…Note that each of Γ 0 and Γ 2 has exactly k edges colored by m + 1 and therefore Γ 0 N = Γ 0 N and Γ 2 N = Γ 2 N . Using equations (2), (3) and (6) is invariant under all Reidemeister moves and is a re-normalization of the ReshetikhinTuraev sl(N ) polynomial for links colored by non-negative integers.…”

Section: Moy Graphsmentioning

confidence: 99%

A colored sl(N) homology for links in S³

2014

Dissertationes Math.

121

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Abstract. Fix an integer N ≥ 2. To each diagram of a link colored by 1, . . . , N , we associate a chain complex of graded matrix factorizations. We prove that the homotopy type of this chain complex is invariant under Reidemeister moves. When every component of the link is colored by 1, this chain complex is isomorphic to the chain complex defined by Khovanov and Rozansky in [19]. The homology of this chain complex decategorifies to the Reshetikhin-Turaev sl(N ) polynomial of links colored by exterior powers of the defining representation.

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“…Therefore the present paper complements these results, giving hope for an eventual complete resolution of the question of when this construction results in a slice knot or link. Our result also provides a method of producing interesting examples for testing the new obstructions to a knot or link being smoothly slice [OS03,Ra04,MO05,BW05,GRS07].…”

Section: Conjecture 11 the Whitehead Double Of A Link Is Freely Slimentioning

confidence: 99%