  Kauffman and Lomonaco explored the idea of understanding quantum entanglement
(the non-local correlation of certain properties of particles) topologically by
viewing unitary entangling operators as braiding operators. In the work of G.
Alagic, M. Jarret, and S. Jordan it is shown that entanglement is a necessary
condition for forming invariants of knots from braid closures via solutions to
the Yang-Baxter Equation. We show that the arguments used by these authors
generalize to essentially the same results for quantum invariant state
summation models of knots. We also given an example of an SU(2) representation
of the three-strand braid group that models the Jones polynomial for closures
of three-strand braids. This invariant is a quantum model for the Jones
polynomial restricted to three strand braids, and it does not involve quantum
entanglement. These relationships between topological braiding and quantum
entanglement can be used as a framework for future work in understanding the
properties of entangling gates in topological quantum computing.The paper ends
with a discussion of the Aravind hypothesis about the direct relationship of
knots and quantum entanglement, and the ER = EPR hypothesis about the
relationship of quantum entanglement with the connectivity of space. We
describe how, given a background space and a quantum tensor network, to
construct a new topological space, that welds the network and the background
space together. This construction embodies the principle that quantum
entanglement and topological connectivity are intimately related.
