🚀 go-pugleaf

Thanks very much, all, for the responses to my "Knotty Problems."  The
returns haven't all come in yet, but if anyone who wrote me would like
to know what everyone *else* had to say about those questions, just
email me.  

Okay, now for something slightly different.   Just as for every group
there is a group algebra, for every monoid there is a monoid algebra.
As far as I recall, the Temperley-Lieb algebra is the monoid algebra
based on the monoid of "tangles on n strands"; which is an extension of
the concept of braids where in addition to crossings:


\   /
 \ /
  \
 / \
/   \


and their inverses, one has things like


\   /
 \_/

  _
 / \
/   \


which don't have inverses.  These satisfy all sorts of charming
relations.  I may have my terminology mixed up a bit here, but I *don't*
need more information on the Temperley-Lieb algebra per se because I
have lots of references on it.  What I'm looking for is somewhat
different.  Namely, what do people call the algebra corresponding to the
monoid generated by crossings, elements like

\   /
 \_/
        
and elements like

  _
 / \
/   \  

Of course the latter two do not conserve the number of strands.

I suppose one could call this simply an "unoriented tangle algebra".
What precisely are the relations for it?  How about the analog for
oriented tangles?  It comes up in the Reshetikhin-Turaev
approach to constructing knot, link, and tangle invariants from quantum
groups (or rigid braided tensor categories).  But are there any good
references on "tangle monoids" and tangle monoid algebras per se?