Do there exist surfaces whose boundary is a closed knot?

In summary, there is a connection between Stoke's theorem and the existence of orientable surfaces. A specific type of surface called a Seifert surface has been linked to this theorem. Additionally, a mathematician named Lawson has shown that all compact orientable surfaces can be realized as minimal surfaces in the 3-sphere, except for the projective plane.
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jk22
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I ask this for the condition of application of Stoke's theorem.
 
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  • #2
Every knot is the boundary of an orientable (you want this for Stokes' theorem) surface. Look up "Seifert surface".
 
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Those slides mention that Lawson, in his dissertation (Annals of Mathematics, November 1970), proved that "all genera" can be realized as minimal surfaces in the 3-sphere.

In fact, Lawson shows that all compact orientable surfaces exist as embedded minimal surfaces in S3, and that all compact non-orientable surfaces but one can occur as immersed minimal surfaces in S3. The exception is the projective plane, which Lawson proves cannot occur as a minimal immersed surface in S3.
 

1. What is a closed knot?

A closed knot is a mathematical concept that refers to a continuous loop formed by a curve or line in three-dimensional space. It can also be thought of as a closed loop with no loose ends or endpoints.

2. What is the boundary of a surface?

The boundary of a surface is the edge or border that separates the surface from its surroundings. In the case of a closed knot, the boundary would be the knot itself.

3. Can a surface have a closed knot as its boundary?

Yes, it is possible for a surface to have a closed knot as its boundary. This is a common topic of study in topology, which is the branch of mathematics that deals with the properties of geometric objects that are preserved through continuous deformations.

4. What are some examples of surfaces with closed knots as boundaries?

One example is a Möbius strip, which is a one-sided surface with a closed knot as its boundary. Another example is a torus, which is a doughnut-shaped surface with a closed knot as its boundary.

5. Why is the study of surfaces with closed knots as boundaries important?

The study of these surfaces has important applications in various fields such as physics, biology, and engineering. It also helps us understand the fundamental properties of surfaces and their boundaries, which can lead to new discoveries and advancements in mathematics and other sciences.

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