I suppose one who could actually stick his head up his ass would be a Klein bottle.
The whole skin is a membrane.
I don’t know what that means, just thought of it…
Waitasec. Your eye sockets are definite bordered cut out holes in your integument. So those are holes in the topology of the skin surface, right?
What does “orientable” mean?
We say ‘asianable’ now.
I assume you are talking about men only, because women definitely have anther hole you aren’t considering.
Which is a dead end, so it doesn’t change the topology.
See I was confused. The OP said both holes and tracts. If we are just talking tracts, then my comment it irrelevant.
Topology is not concerned with “dents” in a structure. Take a flat rubber sheet and distort it by poking, but not breaking it. You have not changed the topology.
Poke a hole in it, however, and you have changed the sheet into a donut, topological-wise. A human body considered as a hole in, connected to a hole out, is topologically equivalent to a donut.
Yeah, it’s kinda weird, but that’s topology.
So you’re saying we’re a hairy donut?
So we should also be this (and then we’re famous).
I’m no anatomist but doesn’t the eye socket close off? Only the optic nerve enters the head. So there’s a connection but no orifice that enters the body.
I’d say the ears are the same, you can go in a little ways but the eardrums seal the inside off from the outside.
I have no idea how the tears ducts fit into this scheme.
So topologically, the human body is equivalent to a sphere with four tubes (anus, mouth, and two nostrils) branching off from a common inner point to the surface.
Probably thirty years ago at least, my SIL was reading some weird new-agey philosophy book where human beings were abstracted as “tubes.” I never read it so I don’t have much else to go on. Anyone have any ideas?
I guess the relevant void has mostly closed up, but I used to be able to hold my nose closed and exhale around one of my eyeballs. I’m not sure if I was forcing air through the tear duct or something else. But yeah, at least genus 5, what with mouth, 2 nostrils, 2 tear ducts, and anus.
But that’s a gross oversimplification. There are bones with holes in them, and so on. The human body isn’t really a single topological solid, even if we pretend those don’t only really exist as abstractions.
This seems strangely relevant:
It means that you can define a “canonical unit normal vector function” on the entire surface in such a way that that function is smooth.
Basically, at each point on the surface, there are two unit vectors that are at right angles (a.k.a. normal) to the surface: One is pointing in the opposite direction to the other one. Pick a point on the surface, and pick one of the unit normal vectors at random as “canonical”. For points near that point, define the normal vector that is closest to the canonical normal vector you’ve already picked to be the “canonical” one. Then do the same at points near those points, until you’ve picked a canonical unit normal for the entire surface.
If at no point on the surface does the canonical unit vector “jump” from one direction to the other at neighboring points, the surface is orientable. If it does, then it is non-orientable.
Classic non-orientable surfaces are the Mobius strip and the Klein bottle. Classic orientable surfaces are pretty much any surface you’d think of when you’re not specifically trying to think of a non-orientable surface. 
There are also the abdominal and thoracic cavities. I don’t precisely know to describe their topology, but they would sort of be like a pair of ring-shaped cavities embedded in a torus such that they encircle the central hole. The usual description I recall from introductions to animal body plans is “tube within a tube”, though we also have a diaphragm that splits up the body cavity.
Also, you can consider the female reproductive tract as a passage into the abdominal cavity, since the end of the fallopian tubes are open to the body cavity.
Hmm do cavities count as holes? Is a hollow shape the same as a donut?
a sprinkle dent in the donut frosting is not the same as a donut hole.
a coffee cup is the same as a donut (don’t confuse in real life, though they still are related there sometimes).
To clarify, a coffee *mug *is a torus because of the hole in the handle. A styrofoam coffee cup is no different than a sheet of paper.
See, now we’re talking volumes.
And this is why I kinda feel this whole “the human body is topographically a torus or a coffee mug!” quite a bit of an oversimplification—and only true in the most broadest of terms.
Our bodies are an overwhelming amalgam of thousands of organs, structures, cavities, hair, pours, and a mind-boggling network of, well, tubes… made up of trillions of cells all pushing solids, fluids and molecules. Sure, it’s all bound by connective tissue, but we’re hardly a homogenous abstract entity like a mathematically idealized, geometric toroidal surface.
I think I read that the eye is the only part of the brain that is visible. Or something.
I wasn’t even thinking of the orifice. Simply, isn’t a sheet of paper with two holes in it a different topological object than a plain sheet?