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Quantum Topology

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Volume 10, Issue 1, 2019, pp. 1–75
DOI: 10.4171/QT/122

Published online: 2018-10-31

A polynomial action on colored $\mathfrak {sl}_2$ link homology

Matthew Hogancamp[1]

(1) University of Southern California, Los Angeles, USA

We construct an action of a polynomial ring on the colored sl2 link homology of Cooper–Krushkal, over which this homology is finitely generated. We define a new, related link homology which is finite dimensional, extends to tangles, and categorifies a scalar multiple of the $\mathfrak {sl}_2$ Reshetikhin–Turaev invariant. We expect this homology to be functorial under 4-dimensional cobordisms. The polynomial action is related to a conjecture of Gorsky–Oblomkov–Rasmussen–Shende on the stable Khovanov homology of torus knots, and as an application we obtain a weak version of this conjecture. A key new ingredient is the construction of a bounded chain complex which categorifies a scalar multiple of the Jones–Wenzl projector, in which the denominators have been cleared.

Keywords: Khovanov homology, Jones–Wenzl idempotent, categorification

Hogancamp Matthew: A polynomial action on colored $\mathfrak {sl}_2$ link homology. Quantum Topol. 10 (2019), 1-75. doi: 10.4171/QT/122