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Braid group representations arising from quantum supergroups with arbitrary q and link polynomials

The theory of quantum supergroups is applied to the construction of braid group representations and link invariants. A method developed in an earlier publication is reviewed, which produces link polynomials from braid group representations generated by quantum supergroup invariant R matrices. When q is a generic complex parameter, this method, applied to the universal R matrices, yields hierarchies of link polynomials associated with different irreps of quantum supergroups. When q is a root of unity, suitable constraints are imposed on quantum supergroups to turn them into finite‐dimensional quasitriangular Z2‐graded Hopf algebras. The corresponding universal R matrices are obtained, which, in turn, lead to series of link polynomials upon using the above mentioned method. As concrete examples, the link polynomials related to the vector irreps of the quantum supergroups Uq(osp(M/2n)), all the irreps of Uq(osp(1/2)) with generic q, and a class of doubly atypical irreps of Uq(gl(2/1)) with q a root of unity...