Knots and Surfaces

Surfaces With Boundary

In class, we have mainly studied surfaces without boundary - the sphere, torus, klein bottle, etc. We have been particularly interested in the Euler characteristics of these surfaces.

How do we find the Euler characteristic of a surface with boundary? Any surface with boundary can be "capped off" to create a surface without boundary simply by filling each boundary component with a disk. Since a disk has Euler characteristic 1, the Euler characteristic of a surface without boundary is equal to the Euler characteristic of the surface obtained by capping it off, minus the number of boundary components in the original surface.

The Euler characteristic of a surface with boundary does not uniquely specify the surface. That is, two non-homeomorphic surfaces may have the same Euler characteristic. Here are two non-homeomorphic surfaces, both with Euler characteristic zero.

So what is different about these surfaces? Notice that the surface on the left is orientable, and the surface on the right is not.

In fact, in order to uniquely determine the homeomorphic type of a surface with boundary, we need know only the following three pieces of information:

Since the two surfaces above both have Euler characteristic zero and two boundary components, we see that they are simply the cylinder and Mobius band, respectively.

Seifert Surfaces

From the perspective of knot theory, some of the more interesting bounded surfaces are those with a single boundary component where the boundary forms a knot. Here are two examples:

The surface on the left is a Mobius band. The surface on the right is a torus minus a disk. Each surface has the trefoil knot as boundary.

In 1934, the German mathematician Herbert Seifert invented an algorithm which, given any knot, will produce an orientable surface with one boundary component such that the boundary component is the knot. This surface is called a Seifert surface for K. Here is the algorithm:

  1. Given a knot K and a projection of that knot, introduce an orientation on K.

  2. At each crossing of the projection, there will be two incoming and two outgoing strands. Connect each incoming strand to the adjacent outgoing strand, thus eliminating crossings, and creating a set of topological circles (Seifert circles) in the plane.

  3. Now, move the circles out of the plane so they are at different heights and fill each one with a disk.

    (looking from the side)

  4. Finally, connect the disks with a series of twisted bands so that, when viewed from the top, the boundaries of the bands look like the original projection of the knot. The resulting surface has one boundary component, and it is the knot K.

We show that any Siefert surface is orientable as follows: Pick two colors, A and B. For each Siefert circle, paint the top Color A and the bottom Color B if the orientation is clockwise, and vice versa if the orientation is counterclockwise. We must show that after adding the connecting bands, this coloration is consistent. Note in every case where two Seifert circles meet, the local appearance is as shown.

There are two ways to connect the ends of the lines, one which yields adjacent circles in the same plane, and one which yields concentric circles, which are moved into different planes by the algorithm.

If the circles are adjacent, one has clockwise orientation and one has counterclockwise orientation, so their tops will be painted different colors. When we add the connecting band, the coloration is consistent. If the circles are concentric, they both have the same orientation, and again when we add the connecting band, the coloration is consistent. Therefore we have successfully painted two sides of the surface different colors, so it must be orientable.

Siefert's algorithm may yield different surfaces for the same knot if different projections of the knot are used. For example, here are the Seifert surfaces generated by two projections of the trefoil knot:

The surfaces formed from different projections will not necessarily have the same genus. However,

Theorem: When Seifert's algorithm is applied to an alternating projection of an alternating knot, the resulting Seifert surface has minimal genus.

The genus of a knot K, written g(K), is defined to be the minimum genus of any Seifert surface for that knot. Here is another theorem, whose proof uses Seifert surfaces:

Theorem: g(K#L) = g(K) + g(L), where K#L is the composition of K and L.

The proof is not difficult to understand, but it is somewhat lengthy and requires some complicated diagrams, so it is not included here. See [Adams] for the proof.

This is a nice theorem because it allows us to prove that the unknot cannot be formed by composing two nontrivial knots. Since the unknot always bounds a disk, its genus is zero. Any nontrivial knot must have genus at least 1, or it would be trivial. So the composition of two nontrivial knots must have genus at least 2, and therefore cannot be the unknot.

Some excercises relating to knots and surfaces:

This page was created using information from chapter 4 of The Knot Book, by Colin Adams, New York, W. H. Freeman and Co., 1994.

If you found this page interesting, you might also enjoy my exploration of intersecting curves.

Sharon Goldwater

Last modified: Tue Mar 18 1997