Tangle Invariants Valued in Derived Categories of Sheaves and the Rouquier Complex

Abstract

We develop effective calculation tools for constructing categorified invariants of tangles and knots, following the work of Cautis and Kamnitzer. These invariants, originally constructed by means of Fourier-Mukai transform functors between derived categories of coherent sheaves on certain iterated flag varieties, can be given an alternative algebraic description by using tilting bundles. We use this method and provide several examples of its application, in the hope of showing its utility to explicit pen-and-paper computations. This tilting-bundle method also allows us to lift the construction from derived to homotopy categories of complexes of modules, reducing the calculation of tangle invariants to a type of categorified matrix algebra, where the matrices in question have maps between their elements and calculations get turbocharged with chain homotopy.