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The cyclotomic Birman-Murakami-Wenzl (BMW) algebras B_n^k, introduced by R.H\"aring-Oldenburg, are a generalisation of the BMW algebras associated withthe cyclotomic Hecke algebras of type G(k,1,n) (aka Ariki-Koike algebras) andtype B knot theory.  In this paper, we prove the algebra is free and of rank k^n (2n-1)!! overground rings with parameters satisfying so-called "admissibility conditions".These conditions are necessary in order for these results to hold andoriginally arise from the representation theory of B_2^k, which is analysed bythe authors in a previous paper. Furthermore, we obtain a geometric realisationof B_n^k as a cyclotomic version of the Kauffman tangle algebra, in terms ofaffine n-tangles in the solid torus, and produce explicit bases that may bedescribed both algebraically and diagrammatically.  The admissibility conditions are the most general offered in the literaturefor which these results hold; they are necessary and sufficient for all resultsfor general n.

On the freeness of the cyclotomic BMW algebras: admissibility and an isomorphism with the cyclotomic Kauffman tangle algebras