When I ask Mathematica to give me the Euler characteristic of
MengerMesh[1, 3], it returns 16. It seems to me the 1-step Menger sponge has genus 5 and should therefore have Euler characteristic -8. For
MengerMesh[2, 3], I'm again off by a factor of -2, as Mathematica returns 320 while I predict -160 based on its genus, which is 81. For
MengerMesh[0, 3], me and Mathematica agree.
Can anyone please explain to me what I'm missing here?