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

Suppose V is the unit disc D embedded in the d-dimensional unit ball Bd and attached to the unit sphere. Consider the space HV, the restriction of the Drury–Arveson space to the variety V, and its multiplier algebra MV=Mult(HV). The isomorphism problem is the following: Is V1≅V2 equivalent to MV1≅MV2? A theorem of Alpay, Putinar and Vinnikov states that for V without self-crossings on the boundary MV 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 MV1≅MV2 algebraically, then V1 and V2 must have the same self-crossings up to a unit disc automorphism. We prove that an isomorphism between MV1 and MV2 can only be given by a composition with a map from V1 to V2. 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.