Mixed quantum skew Howe duality and link invariants of type A

We define a ribbon category Sp(β), depending on a parameter β, which encompasses Cautis, Kamnitzer and Morrison's spider category, and describes for β=m−n the monoidal category of representations of Uq(glm|n) generated by exterior powers of the vector representation and their duals. We identify this category Sp(β) with a direct limit of quotients of a dual idempotented quantum group U˙q(glr+s), proving a mixed version of skew Howe duality in which exterior powers and their duals appear at the same time. We show that the category Sp(β) gives a unified natural setting for defining the colored glm|n link invariant (for β=m−n) and the colored HOMFLY-PT polynomial (for β generic).