Is there any regular polyhedron that is not of Euler characteristic 2

Question

Is there any regular polyhedron that 1. consist of congruent regular polygons as its faces 2. each vertex has same number of adjacent edges

but nonetheless not of characteristic 2? (say, torus or genus?)

If there is any, I would be grateful if you give me its visualization.

Thanks in advance.

Answer 1

The five Platonic solids have Euler characteristic $2$ (we are talking about the boundary). The four Kepler-Poinsot polyhedra do not all share this property. Two have Euler characteristic $-6$, and two have $2$ like the Platonic solids.

Indeed, nonconvex uniform polyhedra can have a variety of values for the Euler characteristic.