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.)

Knot Theory. As such, it is very much studied, and very rich, and very complicated. $\endgroup$ – Theo Johnson-Freyd Mar 28 '11 at 16:471more comment