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 :-)


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.)

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    $\begingroup$ One presentation of TL is as the free undirected planar algebra in which the empty circle can be removed for multiplication by some scalar. Are you proposing anything deeper than the algebra of (framed) tangles in which an unlinked unknot can be removed for multiplication by a scalar (and framing changes can be made for some other scalar)? If so, then your algebra is called Knot Theory. As such, it is very much studied, and very rich, and very complicated. $\endgroup$ – Theo Johnson-Freyd Mar 28 '11 at 16:47
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    $\begingroup$ You might want to look at the (type A) Hecke algebra or the BMW algebra. These have generators similar to what you describe but impose additional relations. One problem with your example is that the morphism spaces are infinite dimensional and difficult to analyze. $\endgroup$ – Kevin Walker Mar 29 '11 at 0:36
  • $\begingroup$ @Theo - yes, I came up with this as a kind of reformulation of knot theory. But the "solution set" (if you interpret S and H as matrices, i.e. in the abstract tensor framework) of polynomial invariants is different: abstract tensor € Narf algebra € knot theory. (As it seems, already Kauffmans 2 variable polynomial is incompatible with abstract tensors. Or so I heard. The intent of all this was to "weaken the whirl move" in a plausible way. BTW, the K2VP is still incompatible.) @Kevin - are the additional relations compatible with knot theory? (I didn't list all of mine, they are.) Hauke $\endgroup$ – Hauke Reddmann Mar 29 '11 at 9:34
  • $\begingroup$ @Kevin - I looked up the original BW paper - the pictures are the same, but an additional relation R+S=c(H+I) is imposed. Since the relations stay the same, I think I can simply make a reference to BW in my paper. Question declared solved. $\endgroup$ – Hauke Reddmann Mar 30 '11 at 8:51
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    $\begingroup$ Are you just looking at the algebra of balanced tangles, with equal number of endpoints at the top and bottom? Also, I don't understand the relation you call "whirl". What is R2? $\endgroup$ – Dylan Thurston Apr 2 '11 at 19:47

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.

  • $\begingroup$ THX. I am out of fashion either so don't worry :-) Could you give me a standard reference and make me a happy panda? (That is, until I try to understand a word of it...Unfortunately, I'm just an amateur. Thus: The more diagrams, the merrier.) $\endgroup$ – Hauke Reddmann May 6 '11 at 8:52

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