## Knots, groups, subfactors and physics.(English)Zbl 1361.46048

This article is based on the 15th Takagi Lectures that the author delivered at Tohoku University on June 27 and 28, 2015. Aimed at an audience of non-specialists, it provides the background and a coherent narrative for the author’s ongoing programme to find invariants for knots and links by geometric constructions involving representations of braid groups and Thompson groups. Giving many instructive details in the less technical parts and putting the more technical parts into context (and providing references), this is a roadmap as well as an invitation.
The contents are as follows. Section 1 gives a short introduction to knot theory and its history, with emphasis on the connections with groups. Section 2 is about braid groups and how to get knots and links from braids. Section 3 describes the Thompson groups $$F$$ and $$T$$ and the more recent construction how to get knots and links from elements of Thompson groups. Section 4 provides an introduction to von Neumann algebras and factors, starting from the basics and leading up to standard forms and module dimensions (the Murray-von Neumann coupling constant). Section 5 about subfactors starts with a warmup about subfactors constructed from outer actions of finite groups and proceeds to a collection of more complicated examples, the basic construction, index of subfactors and the restrictions for the index, the basic construction in finite dimensions as a tool to realise subfactors. Section 6 introduces planar algebras, without giving many technical details but relying on the Temperley-Lieb and Conway algebras as examples. This allows in the final Section 7 to sketch the construction of knots and links from representations of braid groups and Thompson groups in the Conway planar algebra. In principle, this yields a systematic way to get invariants by finding suitable relations and the corresponding quotients of the Conway algebra (referred to as ‘invariants by machine’). The Jones polynomial can be considered as an example. In the case of subfactor planar algebras, we can construct Hilbert spaces and unitary representations. Here the tale becomes more and more sketchy and the interested reader who accepts the invitation now has to go on with the references given.

### MSC:

 46L37 Subfactors and their classification 46L54 Free probability and free operator algebras 57M25 Knots and links in the $$3$$-sphere (MSC2010)

### Keywords:

knots; braid group; Thompson group; subfactors; planar algebras
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### References:

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