I am not sure, but the three circles I described cannot be moved continuously to one another or shrunk to a point. TBH, it surprised me too. The mathematical subject involved is called homotopy theory and I am not an expert in it.
Then again, though, if I start with an ordinary sheet of paper and use a hole-puncher on it four times, then I can draw a whole bunch of independent loops on the paper, far more than 5. At least 16, since I can draw loops that encompass any subset of the punched holes, except it’s even more, since I can wrap those loops around the other punched holes in various ways.
Sounds like knots.
Ok, assuming it’s not an unsolvable trick like “oh there’s a cardboard insert so you can’t see the holes in the back”.
Topologically I suppose my question is how many “holes” does a garbage bag have? Because it’s the same topological shape as a T-shirt. Since we used to make garbage bags into summer camp by cutting 3 holes in them, the answer to the question is either 15 or 16 depending on whether the bottom counts.
Sounds like the fundamental group. Now, in knot theory, the “knot group” is just the fundamental group of what is left after cutting out the knot.
And now it may be worth mentioning that there is a topological theorem that if X is a space, let us say path-connected, then there is a natural map from the fundamental group of X, modulo its commutator subgroup, to H_1(X) which is an isomorphism. So you can draw a bunch of independent loops, but a loop like aba^{-1}b^{-1} is homologous to zero. The homology class of a closed chain is just determined by how much it winds around each of the holes.
The answer is zero.
Ceci n’est pas une chemise.
If I dig a hole in the ground, is it a hole?
I guess it depends on the definition of “hole.”
Does a sock have a hole?
Just remember, a topologist is a guy who can’t tell his ass from a hole in the ground, but can tell his ass from two holes in the ground.
Back to the OP, it seems to me that there are two fundamental issues:
1: What counts as a hole?
2: What’s going on with the back of the shirt that we can’t see?
And 1: can be further divided into
1a: How are holes counted mathematically?
1b: Do intentional things count as holes, or only unintentional holes?
The topology is clear: no.
Whether the problem is strictly about topology or is colloquial arm-waving? Aye, there’s the rub.
Only if it goes through to China.