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Braiding operators corresponding to the third Reidemeister move in the theoryof knots and links are realized in terms of parametrized unitary matrices forall dimensions. Two distinct classes are considered. Their (non-local) unitaryactions on separable pure product states of three identical subsystems (say,the spin projections of three particles) are explicitly evaluated for alldimensions. This, for our classes, is shown to generate entangled superpositionof four terms in the base space. The 3-body and 2-body entanglements (in three2-body subsystems), the 3-tangles and 2-tangles are explicitly evaluated foreach class. For our matrices, these are parametrized. Varying parameters theycan be made to sweep over the domain (0,1).Thus braiding operatorscorresponding to over- and under-crossings of three braids and, on closingends, to topologically entangled Borromean rings are shown, in another context,to generate quantum entanglements. For higher dimensions, starting withdifferent initial triplets one can entangle by turns, each state with all therest. A specific coupling of three angular momenta is briefly discussed tothrow more light on three body entanglements.

Unitary Braid Matrices: Bridge between Topological and Quantum Entanglements