snapPy Simple Example

Question

I'm trying to learn how to use snapPy by doing some examples that I'm familiar with. So, how would I create a genus 1 handleboy and then do a double Dhen filling to create $\mathbb{R}P^3$? Sorry, I tried to figure this out by looking over the documentation but I'm not getting anywhere.

Answer 1

First, I am sure there is an easier way then what I am about to show you. To make $\mathbb{R}P^3$ using only SnapPy, you first can make the unknot using the plink editor. Type:

m=Manifold()

And this automatically opens a window which will let you draw any link. Draw a unknot anyway you want, but I made a triangle. Once it is closed up, go to the dropdown menu "Tools" and select "Send to SnapPy." (There is also a useful "Make Alternating" tool for when you want an alternating link.)

Now you have the manifold that is the complement of the unknot in $S^3$ named $m$. Next type

m.browse()

This will open a new window which has a lot of nice features automatically showing up. We are interested in the "Dehn Filling" section which is on the left side of the window. Since the Lens Space $L(2,1)$ is $\mathbb{R}P^3$, type a "2" into the Meridian area and a 1 in the Longitude area. Then click "Fill." You now have $\mathbb{R}P^3$. You can check any of the invariants which SnapPy has in the browse window. But for this mini example, close the window. Next, to see the fundamental group of $\mathbb{R}P^3$, type:

m.fundamental_group()

and as we expect, it should return:

Generators:
   a
Relators:
   aa

That is basically it. Alternatively, I would suggest using Regina (Made by Ben Burton) instead of SnapPy directly. Regina has a built-in method for converting to SnapPy, so you don't really lose anything here.

Download Regina and open it up. There are some nice tutorials that will help if you want, but if you just want to look at $\mathbb{R}P^3$, click the new 3-manifold triangulation button and it will give two dropdown menus where you can choose "Example Triangulation," from the second one. From there, you get a third dropdown where you can choose $\mathbb{R}P^3$ or whatever manifold you like from their pre-made list. Click Ok.

You should get a new window which has some tabs across the top, the last one is "SnapPea." In this tab, click the only button, "Convert to a SnapPea Triangulation." This opens yet another new window with a lot of SnapPea info. Hope this all helps.