  Closed quantum surfaces of any genus are defined as subalgebras of the
Toeplitz algebra by mimicking the classical construction of identifying arcs on
the boundary of the (quantum) unit disk. Isomorphism classes obtained from
different arrangements of arcs are classified. It is shown that the K-groups
are isomorphic to the classical counterparts and explicit generators of the
C*-algebras and of the K-groups are given.
