[Paper Review] A Categorification of Quantum sl_3 Projectors and the sl_3 Reshetikhin-Turaev Invariant of Tangles

David E. V. Rose|arXiv (Cornell University)|Sep 8, 2011

Algebraic structures and combinatorial models22 references11 citations

TL;DR

This paper constructs a categorification of the quantum sl₃ projectors—key elements in the sl₃ Reshetikhin-Turaev invariant—by taking the stable limit of complexes assigned to k-twist torus braids in Morrison and Nieh's geometric categorification of sl₃ link homology. The stable limit yields a complex that categorifies the sl₃ projector, enabling a full categorification of the sl₃ Reshetikhin-Turaev invariant for framed tangles via a homotopy-invariant assignment of complexes to tangle diagrams with inserted categorified projectors.

ABSTRACT

We construct a categorification of the quantum sl_3 projectors, the sl_3 analog of the Jones-Wenzl projectors, as the stable limit of the complexes assigned to k-twist torus braids (as k goes to infinity) in a suitably shifted version of Morrison and Nieh's geometric formulation of sl_3 link homology (math.GT/0612754). We use these projectors to give a categorification of the sl_3 Reshetikhin-Turaev invariant of framed tangles.

Motivation & Objective

• To extend the categorification of Jones-Wenzl projectors in sl₂ to the sl₃ setting.
• To construct a categorified version of the sl₃ projectors (also called 'internal clasps' or 'magic elements') using a stable limit construction.
• To define a categorified sl₃ Reshetikhin-Turaev invariant for framed tangles that decategorifies to the classical invariant.
• To establish that the resulting invariant is well-defined up to homotopy under Reidemeister moves and projector insertion location.
• To provide a diagrammatic, geometric construction of the categorified projectors using foam calculus and chain complex techniques in the sl₃ spider framework.

Proposed method

• Uses the geometric categorification of sl₃ link homology developed by Morrison and Nieh, which assigns chain complexes to tangles using webs and foams.
• Constructs the categorified sl₃ projector as the stable limit (as k → ∞) of complexes assigned to k-twist torus braids in a suitably shifted complex category.
• Employs homological algebra techniques, including chain complex calculus and Gaussian elimination, to simplify and analyze the complexes arising from tangle diagrams.
• Applies the functor Hom(∅, −) to extract graded vector space cohomology from closed webs, yielding the categorified invariant as a complex of graded vector spaces.
• Uses diagrammatic calculus in the sl₃ spider, including local relations and foam relations, to verify invariance under Reidemeister moves and projector sliding.
• Relies on results from Rozansky on categorified projectors in the sl₂ case as a blueprint, adapting them to the sl₃ setting via the web and foam formalism.

Experimental results

Research questions

• RQ1 Can the quantum sl₃ projectors be categorified in a way analogous to how Jones-Wenzl projectors were categorified in the sl₂ case?
• RQ2 Does the stable limit of complexes assigned to k-twist torus braids in the sl₃ setting yield a well-defined categorified projector?
• RQ3 Is the resulting categorified invariant of framed tangles invariant under R2 and R3 Reidemeister moves and independent of projector insertion location?
• RQ4 Can the categorified sl₃ Reshetikhin-Turaev invariant be constructed via a geometric, diagrammatic formalism using webs and foams?
• RQ5 How do the homological and grading structures of the resulting complexes reflect the classical sl₃ invariant?

Key findings

• The complex assigned to a k-twist torus braid stabilizes as k → ∞, and the stable limit defines a categorified sl₃ projector.
• The categorified projectors satisfy categorified versions of the defining relations of the sl₃ projectors in the spider category.
• The categorified invariant of a framed tangle is well-defined up to homotopy equivalence under R2 and R3 Reidemeister moves.
• The invariant is independent of the location where the categorified projector is inserted along a tangle component.
• Explicit computations show that the cohomology of the invariant complex recovers the classical sl₃ Reshetikhin-Turaev invariants in graded dimensions, with specific grading shifts and ranks.
• For example, the invariant of the (+ + +) tangle yields cohomology groups in gradings j = -4, -2, 0 (rank 1), and higher gradings with periodic structure, matching the expected classical invariant.