  We consider Metropolis-based systematic scan algorithms for generating
Birman-Murakami-Wenzl (BMW) monoid basis elements of the BMW algebra. As the
BMW monoid consists of tangle diagrams, these scanning strategies can be
rephrased as random walks on links and tangles. We translate these walks into
left multiplication operators in the corresponding BMW algebra. Taking this
algebraic perspective enables the use of tools from representation theory to
analyze the walks; in particular, we develop a norm arising from a trace
function on the BMW algebra to analyze the time to stationarity of the walks.
