The Tangle Hypothesis: Dimension 1

Abstract

We introduce an $(\infty ,1)$-category ${\mathsf{Bord}}_1^{\mathsf{fr}}({\mathbb{R}}^n)$, the morphisms in which are framed tangles in ${\mathbb{R}}^n\times {\mathbb{D}}^1$. We prove that ${\mathsf{Bord}}_1^{\mathsf{fr}}({\mathbb{R}}^n)$ has the universal mapping out property of the 1-dimensional Tangle Hypothesis of Baez--Dolan and Hopkins--Lurie: it is the rigid ${\mathcal{E}}_n$-monoidal $(\infty ,1)$-category freely generated by a single object. Applying this theorem to a dualizable object of a braided monoidal $(\infty ,1)$-category gives link invariants, generalizing the Reshetikhin--Turaev invariants.

103 pages, 12 figures