# Does a straw with two branches have 2 holes?

It is easy to show that a standard straw is topologically equivalent to a torus (thus it has one hole). However, what if you had something that was straw-like that had a second "branch" off of it so to speak (like pictured here)? What would this be topologically equivalent to? Does it have 2 holes?

• Yes, your idea is correct.
– MJD
May 13, 2021 at 2:49
• if you considered it as a 2D surface with boundary, it would be topologically equivalent to pants :) May 13, 2021 at 2:57

Yes, it has two holes. Here's one way to think of it. Let's take an ordinary "double torus", like a donut with two holes in it.

Now put some fertilizer on the top part of the torus. The torus will grow upwards, but only near the places where we put the fertilizer. The black ring is where we want to put the fertilizer:

That upward hollow growth is the bottom part of your straw, or the waist part of a pair of cutoff trousers.

We want the growth on the bottom, to match your straw picture, so turn the torus over. Now add fertilizer to the other side:

This time the torus grows upwards in two hollow tubes that are like the sleeves of the trousers or the arms of your straw.

Another way to think about it: a torus has the property that you make a cut all the way through it without cutting the surface into two pieces. On a sphere this is impossible, any cut divides the sphere in two. But on a torus you can do it by having the cut go through the hole:

But you cannot make two such cuts without dividing the torus into at least two pieces.

With a double torus, you can make two cuts without cutting the surface into two pieces, as long as you have one cut go through each hole:

You can make any two of the indicated cuts, and the double torus will stay in one piece. But if you make all three cuts, it falls apart.

Now what kind of thing is your straw? You can indeed make two cuts in it without it falling apart, if you send the two cuts through the two holes:

The blue cut goes down one hole, and the red cut down the other.

You could also have made one of the cuts go down one arm and back up the other arm. The analogous cut on the figure-8-shaped double torus is to cut through the middle section of the figure-8, the purple cut in the double-torus picture above.

As with the double torus, you can make any two of the indicated cuts, and the straw will stay in one piece. But if you make all three, it falls apart: