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arXiv · 2610.17482

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 $\mathsf{GRT}_1$. 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 $\mathsf{GRT}_1$ acts by cyclic automorphisms of parenthesized ribbon chord diagrams.

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Chandan Singh. 2026-10-12. Cyclic reflections and the flip equation for the graded Grothendieck-Teichmüller group. https://arxiv.org/abs/2610.17482

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