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.)