  In this letter, we study the two-spin-1/2 realization for the
Birman-Murakami-Wenzl (B-M-W) algebra and the corresponding Yang-Baxter
$\breve{R}(\theta,\phi)$ matrix. Based on the two-spin-1/2 realization for the
B-M-W algebra, the three-dimensional topological space, which is spanned by
topological basis, is investigated. By means of such topological basis
realization, the four-dimensional Yang-Baxter $\breve{R}(\theta,\phi)$ can be
reduced to Wigner $D^{J}$ function with $J=1$. The entanglement and Berry phase
in the spectral parameter space are also explored. The results show that one
can obtain a set of entangled basis via Yang-Baxter $\breve{R}(\theta,\phi)$
matrix acting on the standard basis, and the entanglement degree is maximum
when the $\breve{R}_{i}(\theta,\phi)$ turns to the braiding operator.
