Grothendieck--Teichmüller Symmetries of Cyclic Operads and Tangles

2025/11/08 by Robertson, Marcy, Singh, Chandan · 1 citation
#18 #18M05 #18M60 #18M75 #55 #Algebraic Topology (math.AT) #Category Theory (math.CT) #FOS: Mathematics #Quantum Algebra (math.QA)

paper · doi:10.48550/arxiv.2511.05911

Abstract

We characterise the profinite Grothendieck-Teichmüller group \widehatGT as the group of automorphisms of the profinite completion of a cyclic operad of parenthesised ribbon braids. This operad generates a symmetric monoidal category which is equivalent to the category of framed, oriented tangles, thereby providing an operadic model for profinite tangles and their arithmetic symmetries. As applications, we show that \widehatGT acts naturally on tangles and provide an alternative proof of the formality of the cyclic framed little disks operad.

Citations

• Models for cyclic infinity operads
• Cyclic operads through modules
• Profinite ∞-operads
• Cyclic framed little disks algebras, Grothendieck-Verdier duality and handlebody group representations
• Profinite completion of operads and the Grothendieck-Teichmüller group
• Galois action on knots I: Action of the absolute Galois group
• Relative categories: Another model for the homotopy theory of homotopy theories
• Cyclic operad formality for compactified moduli spaces of genus zero\n surfaces
• On fusion categories
• Cyclic operads and algebra of chord diagrams
• On Associators and the Grothendieck-Teichmuller Group I
• The universal Vassiliev-Kontsevich invariant for framed oriented links

Cited by

• The Kontsevich invariant and the action of the Grothendieck--Teichmüller group on 2-component string links

Related

• Cyclic framed little disks algebras, Grothendieck-Verdier duality and handlebody group representations
• Profinite completion of operads and the Grothendieck-Teichmüller group
• Racks and blocked braids
• Ribbon Yetter--Drinfeld modules and tangle invariants
• Combinatorial n-fold monoidal categories and n-fold operads