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2009

Denseness and Zariski denseness of Jones braid representations

Kuperberg, Greg

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Using various tools from representation theory and group theory, but without using hard classification theorems such as the classification of finite simple groups, we show that the Jones representations of braid groups are dense in the complex Zariski topology when the parameter $t$ is not a root of unity.

As first established by Freedman, Larsen, and Wang, we the same result when t is a non-lattice root of unity, other than one initial case when t has order 10.
We also compute the real Zariski closure of these representations.
When such a representation is indiscrete in the analytic topology, then its analytic closure is the same as its real Zariski closure.

Denseness and Zariski denseness of Jones braid representations · Around

Built on

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Louis H. Kauffman and Sostenes L. Lins, Temperley-Lieb recoupling theory and invariants of 3-manifolds , Annals of Mathematics Studies, Princeton University Press, Princeton, NJ, 1994
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Similar

Steven Roman, Field theory , Graduate Texts in Mathematics, vol. 158, Springer-Verlag, 1994
1994
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William Fulton and Joseph Harris, Representation theory , third ed., Graduate Texts in Mathematics, vol. 129, Springer-Verlag, New York, 1998
1998
Cited alongside, same era.
Michael H. Freedman, Michael J. Larsen, and Zhenghan Wang, The two-eigenvalue problem and density of Jones representation of braid groups , Comm. Math. Phys. 228
2002
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Dorit Aharonov, Itai Arad, Elad Eban, and Zeph Landau, Polynomial quantum algorithms for additive approximations of the Potts model and other points of the Tutte plane , arXiv:quant-ph/0702008
Cited in the paper.
Greg Kuperberg, How hard is it to approximate the Jones polynomial? , arXiv:0908.0512
Original
Cited in the paper.

Then

E. Breuillard and T. Gelander, On dense free subgroups of Lie groups , J. Algebra 261
2003
Later among the works it cites.
David Witte, Ratner’s theorem on unipotent flows , Chicago Lectures in Mathematics, University of Chicago Press, 2005, arXiv:math/0310402
Original
2005
Later among the works it cites.

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