Type $B$ Webs

We solve the type $B$ case of the main open problem from Kuperberg's 1996 paper "Spiders for rank 2 Lie algebras." That is, we define a $\mathbb{C}(q)$-linear pivotal category $\mathbf{Web}(\mathfrak{so}_{2n+1})$ and prove that it is equivalent to the full subcategory of finite-dimensional representations of $U_q(\mathfrak{so}_{2n+1})$ tensor-generated by the fundamental representations.

A consequence of our main result is an explicit construction of braid group symmetries for nonclassical finite-dimensional representations of the $\iota$quantum group ${U}^{\iota}_{-q^2}(\mathfrak{so}_m)$, which may be of independent interest.