  In this paper, we explore algebraic structures and low dimensional topology
underlying quantum information and computation. We revisit quantum
teleportation from the perspective of the braid group, the symmetric group and
the virtual braid group, and propose the braid teleportation, the teleportation
swapping and the virtual braid teleportation, respectively. Besides, we present
a physical interpretation for the braid teleportation and explain it as a sort
of crossed measurement. On the other hand, we propose the extended
Temperley--Lieb diagrammatical approach to various topics including quantum
teleportation, entanglement swapping, universal quantum computation, quantum
information flow, and etc. The extended Temperley--Lieb diagrammatical rules
are devised to present a diagrammatical representation for the extended
Temperley--Lieb category which is the collection of all the Temperley--Lieb
algebras with local unitary transformations. In this approach, various
descriptions of quantum teleportation are unified in a diagrammatical sense,
universal quantum computation is performed with the help of topological-like
features, and quantum information flow is recast in a correct formulation. In
other words, we propose the extended Temperley--Lieb category as a mathematical
framework to describe quantum information and computation involving maximally
entangled states and local unitary transformations.
