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Abstract. Using totally symmetric sets, Chudnovsky–Kordek–Li–Partin gave a superexponential lower bound on the cardinality of non-abelian finite quotients of the braid group. In this paper, we develop new techniques using multiple totally symmetric sets to count elements in non-abelian finite quotients of the braid group. Using these techniques, we improve the lower bound found by Chudnovsky et al. We exhibit totally symmetric sets in the virtual and welded braid groups and use our new techniques to find superexponential bounds for the finite quotients of the virtual and welded braid groups.
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This content will become publicly available on December 1, 2025
Algebraic weaves and braid varieties
abstract: In this manuscript we study braid varieties, a class of affine algebraic varieties associated to positive braids. Several geometric constructions are presented, including certain torus actions on braid varieties and holomorphic symplectic structures on their respective quotients. We also develop a diagrammatic calculus for correspondences between braid varieties and use these correspondences to obtain interesting decompositions of braid varieties and their quotients. It is shown that the maximal charts of these decompositions are exponential Darboux charts for the holomorphic symplectic structures, and we relate these charts to exact Lagrangian fillings of Legendrian links.
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- Award ID(s):
- 2302305
- PAR ID:
- 10613603
- Publisher / Repository:
- Johns Hopkins University Press
- Date Published:
- Journal Name:
- American Journal of Mathematics
- Volume:
- 146
- Issue:
- 6
- ISSN:
- 1080-6377
- Page Range / eLocation ID:
- 1469 to 1576
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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