The Tangle Hypothesis: Dimension 1

TL;DR

This paper constructs an $( abla,1)$-category of framed tangles in $ ^n$ and proves it satisfies the universal property of the 1-dimensional Tangle Hypothesis, leading to new link invariants.

Contribution

It introduces a new $( abla,1)$-category of framed tangles and proves its universal property, connecting it to the Tangle Hypothesis and link invariants.

Findings

Establishes the universal property of the tangle category

Generalizes Reshetikhin--Turaev link invariants

Provides a new framework for $( abla,1)$-categories

Abstract

We introduce an $(\infty,1)$-category ${\sf Bord}_1^{\sf fr}(\mathbb{R}^n)$, the morphisms in which are framed tangles in $\mathbb{R}^n\times \mathbb{D}^1$. We prove that ${\sf Bord}_1^{\sf fr}(\mathbb{R}^n)$ 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.