The Tangle Hypothesis: Dimension 1

2024/10/31 by David Ayala, Ayala, David, John Francis +1 · 1 voice
Mathematics · #Algebraic structures and combinatorial models #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #math.AT #math.CT #math.GT #math.QA

paper · pdf · doi:10.48550/arxiv.2410.23965

openalex publication_date 2024/10/31 · openalex created_date 2024/11/14 · openalex updated_date 2026/07/28

Abstract

We introduce an (∞,1)-category \sf Bord₁^{\sf fr}(ℝⁿ), the morphisms in which are framed tangles in ℝⁿ× \mathbbD¹. We prove that \sf Bord₁^{\sf fr}(ℝⁿ) has the universal mapping out property of the 1-dimensional Tangle Hypothesis of Baez--Dolan and Hopkins--Lurie: it is the rigid E_n-monoidal (∞,1)-category freely generated by a single object. Applying this theorem to a dualizable object of a braided monoidal (∞,1)-category gives link invariants, generalizing the Reshetikhin--Turaev invariants.

Discussions

• The 1-dimensional tangle hypothesis is a theorem! arxiv.org/abs/2410.23965 [bsky, 10 points, 2 comments]

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