2 added 15 characters in body

The relationship between operator algebras and braids is fairly straightforward to explain, and is nicely written up in many places (e.g. in Kauffman's Knots and Physics). Jones studied representations of the braid group $B_n$ into the Temperley-Lieb algebra $TL_n$. The existence of such a representation is not so surprising (the following explanation is with hindsight- historically, this isn't how it happened): A Temperley-Lieb element is a transfer matrix in a Potts model, in which each $e_i$ implements one more interaction, and you can think of a braid as a motion of $n$ distinct points in the lattice, with the crossings of points as an interaction, so it's roughly sort of like a universal model for this type of lattice statistic mechanical setup. Taking the trace of the representation gives the partition function for the model, so it's a natural thing for a statistical mechanic to do. Diagrammatically, taking the trace may be visualized as closing the braid.

The surprise is that the trace is invariant under Markov moves, and by Markov's theorem, any two braids whose closure gives the same knot are related by a finite sequence of Markov moves. Thus, the trace of the representation actually ends up giving an invariant of a knot!!!

Birman said in a talk that Jones verified invariance under Markov moves by chance- basically it was good luck, and nobody could have anticipated such invariance, or that the Jones polynomial would be a knot invariant. So there is some significant element of mystery in the question of what knots have to do with operator algebras. It ties in to the biggest question in quantum topology, which is a curious one: What do quantum invariants mean topologically? Why is any quantum invariant a topological invariant?

The braid group is a group (tautology alert) which mimics interacting point particleson a lattice, so you could expect imagine that it to might be related to operator algebras (or to subfactors), but a knot isn't an element of a group, and the algebraic structure which you can equip the set of knots with is much more complicated and mysterious.

1

The relationship between operator algebras and braids is fairly straightforward to explain, and is nicely written up in many places (e.g. in Kauffman's Knots and Physics). Jones studied representations of the braid group $B_n$ into the Temperley-Lieb algebra $TL_n$. The existence of such a representation is not so surprising (the following explanation is with hindsight- historically, this isn't how it happened): A Temperley-Lieb element is a transfer matrix in a Potts model, in which each $e_i$ implements one more interaction, and you can think of a braid as a motion of $n$ distinct points in the lattice, with the crossings of points as an interaction, so it's roughly sort of like a universal model for this type of lattice statistic mechanical setup. Taking the trace of the representation gives the partition function for the model, so it's a natural thing for a statistical mechanic to do. Diagrammatically, taking the trace may be visualized as closing the braid.

The surprise is that the trace is invariant under Markov moves, and by Markov's theorem, any two braids whose closure gives the same knot are related by a finite sequence of Markov moves. Thus, the trace of the representation actually ends up giving an invariant of a knot!!!

Birman said in a talk that Jones verified invariance under Markov moves by chance- basically it was good luck, and nobody could have anticipated such invariance, or that the Jones polynomial would be a knot invariant. So there is some significant element of mystery in the question of what knots have to do with operator algebras. It ties in to the biggest question in quantum topology, which is a curious one: What do quantum invariants mean topologically? Why is any quantum invariant a topological invariant?

The braid group is a group (tautology alert) which mimics interacting point particles on a lattice, so you could expect it to be related to operator algebras, but a knot isn't an element of a group, and the algebraic structure which you can equip the set of knots with is much more complicated and mysterious.