Seifert Surfaces

a trefoil knotHello!  I’m Sean, a graduate student here at the University of Texas and an aspiring topologist.  Topologists seem to like pictures as a general rule, and I’m no exception, so I hope to post some nice pictures of interesting mathematical objects.

Speaking of, the picture on the right is a trefoil knot.  (Isn’t it pretty?)  We usually think of such knots as sitting inside S^3 (or \mathbb{R}^3 if you like).

trefoil_seifert_tracedNow it’s a fact that given any knot K, there’s an embedded orientable surface in S^3 whose boundary is K.  Such a surface is called a Seifert surface or a spanning surface.  The picture on the right shows a Seifert surface for the trefoil.  As you can see, it consists of two regions on the left and right joined by three twisted bands.  We can tell that it’s orientable because there’s no way to cross from the “top” to the “bottom” of the surface.  In other words, there are no embedded Mobius bands in the surface.

What kind of surface is this?  We know it has one boundary component and that it’s orientable, so from the classification of orientable surfaces we know that this must be a surface of genus g with one puncture.  To find out the genus of the surface, we can cut along the top two twists, being careful to keep track of how we cut.  Then we can untwist at the bottom and straighten the surface out so that it lies flat.  Finally, we can glue back the pieces marked with arrows as indicated.  Hurray!  Now we recognize the surface as a punctured torus!

trefoil_cut_traced

(Cognoscenti will note that we could have seen immediately  that a once punctured orientable surface with the homotopy type of the wedge of two circles must be a punctured torus.  I think it’s more fun to draw pictures.)

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3 Responses to “Seifert Surfaces”


  1. 1 Shaun April 20, 2009 at 11:06 am

    How does one recognize that the last figure is a punctured torus?

  2. 2 Gilbert Bernstein April 27, 2009 at 6:21 pm

    Hey Sean, How are you doing your drawings?

  3. 3 rsbowman April 30, 2009 at 12:29 pm

    Shaun,

    One way is to start with a torus with a hole in it and gradually enlarge the hole. Another way is to start by viewing the punctured torus as a tube (an annulus) with a band attached connecting the boundary circles. It helps to make drawings!


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