Topological Basis Associated with BWMA, Extremes of L1-norm in Quantum Information and Applications in Physics

The topological basis associated with Birman-Wenzl-Murakami algebra (BWMA) is constructed and the three dimensional forms of braiding matrices S have been found for both $S^+=S$ and $S^+=S^{-1}$. A familiar spin-1 model related to braiding matrix associated with BWMA is discussed. The extreme points $(\theta=\pm\pi/2$ and $\pm\pi)$ of L1-norm and von Neumann entropy are shown to be connected to each other. Through the general discussion and examples we then point out that the L1-norm describes quantum entanglement.