An exceptional collection for Khovanov homology

Abstract

The Temperley–Lieb algebra is a fundamental component of $SU\left(2\right)$ topological quantum field theories. We construct chain complexes corresponding to minimal idempotents in the Temperley–Lieb algebra. Our results apply to the framework which determines Khovanov homology. Consequences of our work include semi-orthogonal decompositions of categorifications of Temperley–Lieb algebras and Postnikov decompositions of all Khovanov tangle invariants.