The isomorphism problem for analytic discs with self-crossings on the boundary

Suppose $V$ is the unit disc $\mathbb{D}$ embedded in the $d$-dimensional unit ball ${\mathbb{B}}_d$ and attached to the unit sphere. Consider the space ${\mathcal{H}}_V$, the restriction of the Drury-Arveson space to the variety $V$, and its multiplier algebra ${\mathcal{M}}_V=\mathrm{Mult}({\mathcal{H}}_V)$. The isomorphism problem is the following: Is $V_1\cong V_2$ equivalent to ${\mathcal{M}}_{V_1}\cong {\mathcal{M}}_{V_2}$? A theorem of Alpay, Putinar and Vinnikov states that for $V$ without self-crossings on the boundary ${\mathcal{M}}_V$ is the space of bounded analytic functions on $V$. We consider what happens when there are self-crossings on the boundary and prove that if ${\mathcal{M}}_{V_1}\cong {\mathcal{M}}_{V_2}$ algebraically, then $V_1$ and $V_2$ must have the same self-crossings up to a unit disc automorphism. We prove that an isomorphism between ${\mathcal{M}}_{V_1}$ and ${\mathcal{M}}_{V_2}$ can only be given by a composition with a map from $V_1$ to $V_2$. In the case of a single simple self-crossing we show that there are only two possible candidates for this map and find these candidates. Finally, we provide a continuum of $V$’s with the same self-crossing pattern such that their multiplier algebras are all mutually non-isomorphic.