Soap-film-like minimal surfaces spanning knots

It is proved that any three-times continuously differentiable, nontrivial knot in three-dimensional euclidean space supports a surface that minimizes area among nearby surfaces but that does not touch all of the supporting knot. This provides a mathematical model of a physical phenomenon occurring in soap-films. The notion of a homotopically spanning surface is defined to determine an appropriate class of admissible surfaces, and it is shown that there is a lower bound on the area of admissible surfaces. The existence of an area minimizing admissible surface is then proved by the direct method based on earlier work of E. R. Reiffenberg.