I would be very astonished if this algebra isn't named. 


You simply have the braid AND the Temperley-Lieb generator in
the algebra. Rules are the usual Reidemeister equivalents
plus the kink and whirl move equivalents (2nd question - 
I call them that way, but are there established names?):


Sn braid generator  

Hn Temperley-Lieb generator  

and now you have obvious relations like  

Sn\*Hn=f\*Hn (f writhe fudge factor) - Reidemeister 1  

H1\*H2\*H1=H1 (kink)  

H1\*H2\*S1=H1\*R2 (whirl)  

and so on. The fun is that since the Temperley-Lieb generator
consists of cup PLUS cap, the rules are a bit weaker than
the one you use for the abstract tensor approach. (Proof:
Take H=tensorprod(id,id). Won't work in usual knot theory.
But trivially in this algebra if S=R=H too.)


So please tell me the official name for pasting it in my great unfinished
novel. I hate to call it Narf algebra due to my great impressedness :-)


Hauke


P.S. The algebra wasn't very useful yet, but if I pull the same stunt
with Kuperbergs G2 invariant and B2 spider, it gets VERY interesting!
(PM for details.)



Braid*Temperley-Lieb=?
This approach where you restrict your attention to algebras consisting only of the kinds of diagrams where there's the same number of strands coming in as coming out is a little bit out of fashion (its heyday was mostly the 80's and early 90's) and so people have mostly stopped naming them. The more typical thing to do now is to allow any number of inputs and any number of outputs. This gives you a different structure called a "tensor category" or (roughly equivalently) a "planar algebra." All of the information of the algebras with "balanced" number of inputs or outputs are still there, but it's packaged together in a more convenient way. For example, there's the BMW algebras (one for each number of strands), but also the BMW (or "Dubrovnik polynomial") tensor category where you allow any number of input strands and any number of output strands. Usually you can just use the same name for the algebras or the tensor category.


In your case you're looking at the category of tangles. The individual algebras themselves would usually just be called the endomorphism algebras in the category of tangles. Dylan's suggestion of the "algebra of balanced tangles" is more concise while still being clear.


