Blowing up an arc

Lines intersect as they pass through a point, but you don’t want them to? Blow it up!

Remove a neighborhood of the point.

And sew back in a punctured projective space.

For a point on a surface like we have here, we sew back in a Mobius band.

Now cross this with $I=[-1,1]$ .

Okay, so we sewed in a twisted $I$ bundle over the punctured projective space. But, c’mon, why not?

Now let’s round it out…

… and reinterpret. We blow up a trivial 1-strand tangle to get a solid torus, the complement of the trivial knot. What happens when we blow up a knotted 1-strand tangle?

Update: One more picture.

~ by Ken Baker on September 14, 2008.

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5 Responses to “Blowing up an arc”

This is easily the best visualization of what blowing up does. I can just barely wrap my brain around it without the pictures. Too bad you can’t show a blowup of a point in 3-space.

John Armstrong said this on September 14, 2008 at 12:32 am | Reply
Sounds like a challenge…

Ken Baker said this on September 14, 2008 at 4:38 pm | Reply
These are extremely nice pictures! Which program did you use to draw these?

Sirof said this on October 18, 2008 at 12:19 pm | Reply
Hi Sirof. These were drawn using Rhino3D. (Actually it’s a pre-beta version for the Mac that I’m using. They’re in the process of porting it.) Rhino as well as other 3D modelers these days have functions that allow you to deform and distort objects with control. So for the thickened Mobius band, I made a thickened rectangle, put in a 180 twist, and then bent it around a circle.

Other pictures on this blog are done with SketchUp.

Ken Baker said this on October 26, 2008 at 10:07 pm | Reply
[…] This post, for example, perfectly illustrates the notion of blowing up a point on a manifold. […]

Richard Elwes - Sketches of topology said this on February 8, 2009 at 7:05 pm | Reply