The tangle category $\text{Tang}$ is the category whose objects are the non-negative integers $n$ and whose morphisms $n \to m$ are collections of paths from $n$ points arranged on a line in $\mathbb{R}^3$ to $m$ points on another line in $\mathbb{R}^3$ up to ambient isotopy, possibly with some links thrown in. Composition is given by connecting the paths together. In particular, $\text{Hom}(0, 0)$ consists of links in $\mathbb{R}^3$.
$\text{Tang}$ is a monoidal category with the monoidal operation given by disjoint union. It is also braided monoidal with braiding given by the obvious geometric move. Finally, it has duals given by flipping morphisms upside-down, and every object is self-dual. Note that the core of $\text{Tang}$ is the braid category.
Theorem: $\text{Tang}$ is the free braided monoidal category with duals on one self-dual, unframed object.
This fact, and others like it, have some remarkable consequences. For example, they allow us to write down knot and link invariants from braided monoidal categories, prominent examples including the categories of representations of quantum groups; the Jones polynomial arises in this way from the category of representations of $\mathcal{U}_q(\mathfrak{sl}_2)$. There is a fairly thorough exposition in Kassel's Quantum Groups.
