Knot invariants and higher representation theory II: the categorification of quantum knot invariants

By Ben Webster. 2010

61 citations recorded by the source corpus.

We construct knot invariants categorifying the quantum knot variants for all representations of quantum groups. We show that these invariants coincide with previous invariants defined by Khovanov for sl(2) and sl(3) and by Mazorchuk-Stroppel and Sussan for sl(n). Our technique uses categorifications of the tensor product representations of Kac-Moody algebras and quantum groups, constructed a prequel to this paper. These categories are based on the pictorial approach of Khovanov and Lauda. In this paper, we show that these categories are related by functors corresponding to the braiding and (co)evaluation maps between representations of quantum groups. Exactly as these maps can be used to define quantum invariants attached to any tangle, their categorifications can be used to define knot homologies.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/4dba333fe4819d052ac8781998f99eb9580b783f

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Geometry and Topology

1. David E. V. Rose
2. Hoel Queffelec
3. You Qi
4. Mikhail Khovanov
5. Aaron D. Lauda
6. I. Cherednik
7. Alexander P. Ellis
8. Paul Wedrich

In Mathematical Physics

1. David E. V. Rose
2. Hoel Queffelec
3. You Qi
4. Aaron D. Lauda
5. Mikhail Khovanov
6. I. Cherednik
7. Alexander P. Ellis
8. Sabin Cautis

Citations in the corpus

• An introduction to diagrammatic algebra and categorified quantum sl(2) (2011, background): “We hope that by carefully explaining the definition and motivation behind categorified quantum sl2 the reader will gain sufficient insight to understand the categorifications of all quantum groups, and their role in Webster’s approach to categorification of all Reshetikhin-Turaev invariants [68].”
• The Odd nilHecke Algebra and its Diagrammatics (2011, background): “Various steps in this direction have since been completed, most notably by Webster [18, 19].”
• An approach to categorification of some small quantum groups II (2012, methodology): “Webster [Web10a, Web10b] categorified the tensor products of irreducible representations of quantum groups (adapting the geometric picture of Zheng [Zhe08b, Zhe08a]), which are used to construct ReshetikhinTuraev invariants of links.”
• Categorifications from planar diagrammatics (2010, background): “Moreover, they serve as a building block in Webster’s categorification of Reshetikhin–Turaev link and tangle invariants [49, 50 ].”
• Khovanov homology is a skew Howe 2-representation of categorified quantum sl(m) (2012, background): “Categorifying sln link invariants labeled by arbitrary (non-fundamental) representations appears to be a much more difficult problem [73].”
• Jones polynomials of torus knots via DAHA (2011, background): “It is connected with the latest conjectural extension of the Khovanov–Rozansky theory via the categorification of arbitrary irreducible representations of QGs; see [40] and references therein.”