A generalized pretzel knot (or link) is an unoriented link that, up to isotopy, bounds an unoriented, possibly non-orientable surface P(t_1,...,t_k) formed by joining two parallel 2-disks in horizontal planes in R^3 with k bands that have vertical line segments as their core arcs, where the ith band has t_i half-twists. (A classical pretzel has k = 3; if k = 2 you get a torus knot or link of type (2,t_1+t_2); if k = 1 you get an unknot.) I would like an algorithm (if possible) to pass from (t_1,...,t_k) to a closed braid representation of any one of (or all of?) the oriented links that can be obtained by variously orienting the components of the generalized pretzel P(t_1,...,t_k).

An interesting, possibly suggestive, example is P(-3,3,-2). It is a non-orientable surface. Its boundary is the knot 8_20, a ribbon knot. The ribbon disk it bounds corresponds to the "band representation" (my language) (b(1),b(2)) in the 3-string braid group B_3, where b(1)=\sigma_1 and b(2)=(\sigma_2)^3\sigma_1(\sigma_2)^{-3}; in particular, the closure of the braid

\sigma_1(\sigma_2)^3\sigma_1(\sigma_2)^{-3} is 8_20. Furthermore, if you draw the standard closed-braid diagram of that length 8 3-string braid, its non-orientable checkerboard surface is isotopic (in the plane, or on S^2, depending on how you close your braids) to P(-3,3,-2). So in this one case (and of course a passel of very similar ones) there's an algorithm, sort of. But I don't see any way to make it much more general.

Google found me an article by a Polish physicist called, hopefully, "Braids for Pretzel Knots", but it doesn't seem helpful to me (and the most promising among his references don't actually appear to have the content that say what he says they say, although since I don't speak Physics the problem may be mine, not his). Other than that, Google found nothing. So here I am.