The Tangle Hypothesis: Dimension 1

We introduce an $(\mathrm{\infty },1)$-category ${\mathsf{B}\mathsf{o}\mathsf{r}\mathsf{d}}_1^{\mathsf{f}\mathsf{r}}({\mathbb{R}}^n)$, the morphisms in which are framed tangles in ${\mathbb{R}}^n\times {\mathbb{D}}^1$. We prove that ${\mathsf{B}\mathsf{o}\mathsf{r}\mathsf{d}}_1^{\mathsf{f}\mathsf{r}}({\mathbb{R}}^n)$ 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 $(\mathrm{\infty },1)$-category freely generated by a single object. Applying this theorem to a dualizable object of a braided monoidal $(\mathrm{\infty },1)$-category gives link invariants, generalizing the Reshetikhin–Turaev invariants.