Dear topologists of HN, how do I visualize the deformation of a lion made of clay into a donut, if we model that a biologically accurate lion has one entrance orifice for solids and liquids and not one, but two, exit orifices which form a connected cave system inside the lion? (We may ignore nostrils, lungs, skin pores, and all other orifice systems.)
I was looking at the Wikipedia genus article:
https://en.wikipedia.org/wiki/Genus_%28mathematics%29
If my layman's understanding is correct, this means if we start with a clay sphere of genus 0 and bore a hole through the center of it we get genus 1, a torus. If we continue to bore additional holes through the center of the sphere that connect to our existing cave system we never get to genus 2, because with genus 2 you have two separate cave systems, a lump with two holes A and B such that entering hole A to explore it means that you can't explore hole B until you first exit hole A, so I conclude we must remain at genus 1.
But I don't see how you do a continuous deformation of a clay lump with three or more entrances into tunnels that meet in the middle into a donut with only one hole.