Definition By Braid Representation
Jones' original formulation of his polynomial came from his study of operator algebras. In Jones' approach, it resulted from a kind of "trace" of a particular braid representation into an algebra which originally arose while studying certain models, e.g. the Potts model, in statistical mechanics.
Let a link L be given. A theorem of Alexander's states that it is the trace closure of a braid, say with n strands. Now define a representation of the braid group on n strands, Bn, into the Temperley–Lieb algebra TLn with coefficients in and . The standard braid generator is sent to, where are the standard generators of the Temperley–Lieb algebra. It can be checked easily that this defines a representation.
Take the braid word obtained previously from L and compute where tr is the Markov trace. This gives, where is the bracket polynomial. This can be seen by considering, as Kauffman did, the Temperley–Lieb algebra as a particular diagram algebra.
An advantage of this approach is that one can pick similar representations into other algebras, such as the R-matrix representations, leading to "generalized Jones invariants".
Read more about this topic: Jones Polynomial
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