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## "A polynomial invariant for knots via von Neumann algebras", Vaughn Jones, 1985

*Reviewed August 31, 2023*

*Citation:* Jones, Vaughan FR. "A polynomial invariant for knots via von Neumann algebras." (1985): 103-111.

*Web:* https://projecteuclid.org/journals/bulletin-of-the-american-mathematical-society-new-series/volume-12/issue-1/A-polynomial-invariant-for-knots-via-von-Neumann-algebras/bams/1183552338.full

*Tags:* Foundational, Mathematical, Subfactors

This is the article which introduces the Jones polynomial.

The paper is very short - it does what it sets of to do, without much funny business.
The invariant is defined in terms of subfactors - the modern simplified treatment had to wait until a work of Kauffman

> Kauffman, Louis H. "State models and the Jones polynomial." Topology 26.3 (1987): 395-407.

Interestingly enough, Jones decided to use the language of braids and Markov moves instead of links and Reidemeister moves.
Given any braid, one can "close" it by connecting the top and bottom of every spot on the braid.
This induces a surjection from braids to links, by a theorem of Alexander.
The question is when two braids give the same link - exactly when they are related by Markov moves.
Jones says that his invariant now gives a powerful method for telling apart (almost) any knot.

Another fun thing I got from reading this paper: apparently Jones was already aware of the HOMFLY polynomial when writing his original work!
There is a note at the end of the paper, which was added during the proofreading process: "the similarity between the relation of Theorem 12
and Conway's relation has led several authors to a two-variable generalization of [the Jones polynomial]- This has been done (independently) by Lickorish and Millett, Ocneanu, Freyd and Yetter, and Hoste".
Their paper was published later the same year:

> Freyd, Peter, et al. "A new polynomial invariant of knots and links." (1985): 239-246.