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We establish isomorphisms between certain specializations ofBirman-Murakami-Wenzl algebras and the symmetric squares of Temperley-Liebalgebras. These isomorphisms imply a link-polynomial identity due to W. B. R.Lickorish. As an application, we compute the closed images of the irreduciblebraid group representations factoring over these specialized BMW algebras.

An algebra-level version of a link-polynomial identity of Lickorish