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PreprintCC BY 4.0arXiv

Cyclic reflections and the flip equation for the graded Grothendieck-Teichmüller group

  • Chandan Singh
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Abstract

Furusho, Hirose and Komiyama reformulated the pentagon equation using an involution called the flip, which led to a new characterisation of the graded Grothendieck-Teichmüller group GRT1. We show that the flip is a reflection in the cyclic structure of the operad of framed Drinfeld-Kohno Lie algebras, with the framings removed. Applying the reflection twice to the pentagon gives the two-cycle relation for every Drinfeld associator. This relation is precisely the condition for cyclic compatibility on the associativity morphism, giving a new proof that GRT1 acts by cyclic automorphisms of parenthesized ribbon chord diagrams.

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Cite

BibTeXRIS
BibTeX
@misc{singh2026cyclic,
  title = {{Cyclic reflections and the flip equation for the graded Grothendieck-Teichmüller group}},
  author = {Chandan Singh},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.17482},
  eprint = {2610.17482},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.17482},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Cyclic reflections and the flip equation for the graded Grothendieck-Teichmüller group
AU  - Chandan Singh
PY  - 2026
DA  - 2026/10/12
PB  - arXiv
DO  - 10.48550/arxiv.2610.17482
UR  - https://arxiv.org/abs/2610.17482
AB  - Furusho, Hirose and Komiyama reformulated the pentagon equation using an involution called the flip, which led to a new characterisation of the graded Grothendieck-Teichmüller group GRT_1. We show that the flip is a reflection in the cyclic structure of the operad of framed Drinfeld-Kohno Lie algebras, with the framings removed. Applying the reflection twice to the pentagon gives the two-cycle relation for every Drinfeld associator. This relation is precisely the condition for cyclic compatibility on the associativity morphism, giving a new proof that GRT_1 acts by cyclic automorphisms of parenthesized ribbon chord diagrams.
ER  -