On the Uq(osp(1|2n)) and U₋q(so(2n+1)) Uncoloured Quantum Link Invariants

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Let L be a link and Φᵃₗ(q) its link invariant associated with the vector representation of the quantum (super)algebra Uq(A). Let Fₗ(r,s) be the Kauffman link invariant for L associated with the Birman--Wenzl--Murakami algebra BWMf(r,s) for complex parameters r and s and a sufficiently large rank f. For an arbitrary link L, we show that Φosp(1|2n)ₗ(q) = Fₗ(-q²ⁿ,q) and Φso(2n+1)ₗ(-q) = Fₗ(q²ⁿ,-q) for each positive integer n and all sufficiently large f, and that Φosp(1|2n)ₗ(q) and Φso(2n+1)ₗ(-q) are identical up to a substitution of variables. For at least one class of links Fₗ(-r,-s) = Fₗ(r,s) implying Φosp(1|2n)ₗ(q) = Φso(2n+1)ₗ(-q) for these links.