Tangle and Brauer diagram algebras of type Dn

A generalization of the Kauffman tangle algebra is given for Coxeter type Dn. The tangles involve a pole of order 2. The algebra is shown to be isomorphic to the Birman–Murakami–Wenzl algebra of the same type. This result extends the isomorphism between the two algebras in the classical case, which, in our set-up, occurs when the Coxeter type is An - 1. The proof involves a diagrammatic version of the Brauer algebra of type Dn of which the generalized Temperley–Lieb algebra of type Dn is a subalgebra