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The geometric combinatorics of k-shuffle polynomials

  • Chloé Postel-Vinay
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Abstract

This paper is the second in a series of two which aims at introducing a new family of groups called k-shuffle braid groups, denoted ℬn(k). We focus here on the geometrical origins of these groups, as first observed by Calegari in arXiv:2105.11265. The spaces we are interested in are spaces of complex monic polynomials of degree n whose critical values lie in ℂ ∖ {1, e2πi/k, …, e(2πi (k−1))/k}. We show that the k-shuffle braid groups are the fundamental groups of those spaces, which correspond geometrically to complements of k translates of the discriminant variety. Moreover, we show that the k-shuffle braid groups ℬn(k) inject into the larger braid groups ℬnk.

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Cite

BibTeXRIS
BibTeX
@misc{postelvinay2026geometric,
  title = {{The geometric combinatorics of $k$-shuffle polynomials}},
  author = {Chloé Postel-Vinay},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.33890},
  eprint = {2610.33890},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.33890},
}
RIS
TY  - UNPB
TI  - The geometric combinatorics of k-shuffle polynomials
AU  - Chloé Postel-Vinay
PY  - 2026
DA  - 2026/10/24
PB  - arXiv
DO  - 10.48550/arxiv.2610.33890
UR  - https://arxiv.org/abs/2610.33890
AB  - This paper is the second in a series of two which aims at introducing a new family of groups called k-shuffle braid groups, denoted ℬ_n(k). We focus here on the geometrical origins of these groups, as first observed by Calegari in arXiv:2105.11265. The spaces we are interested in are spaces of complex monic polynomials of degree n whose critical values lie in ℂ ∖ {1, e^2πi/k, …, e^(2πi (k−1))/k}. We show that the k-shuffle braid groups are the fundamental groups of those spaces, which correspond geometrically to complements of k translates of the discriminant variety. Moreover, we show that the k-shuffle braid groups ℬ_n(k) inject into the larger braid groups ℬ_nk.
ER  -