Do there exist surfaces whose boundary is a closed knot?

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Discussion Overview

The discussion centers around the existence of surfaces whose boundary is a closed knot, particularly in the context of applying Stokes' theorem. It explores theoretical implications and mathematical constructs related to knots and surfaces.

Discussion Character

  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant inquires about the conditions for applying Stokes' theorem in relation to knots and surfaces.
  • Another participant asserts that every knot can be the boundary of an orientable surface, referencing Seifert surfaces as a relevant concept.
  • A link to a document is provided, which discusses minimal and Seifert surfaces in relation to knots.
  • A further contribution mentions Lawson's work, indicating that all genera can be realized as minimal surfaces in the 3-sphere, and details the existence of various compact orientable and non-orientable surfaces as embedded or immersed minimal surfaces in S3, with a specific exception noted for the projective plane.

Areas of Agreement / Disagreement

Participants present differing views on the existence and characteristics of surfaces related to knots, with some asserting the existence of orientable surfaces and others providing specific examples and exceptions. The discussion remains unresolved regarding the implications of these findings for Stokes' theorem.

Contextual Notes

The discussion involves complex mathematical concepts that may depend on specific definitions and assumptions regarding surfaces and knots, which are not fully explored or resolved in the thread.

jk22
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I ask this for the condition of application of Stoke's theorem.
 
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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.
 

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