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seminars:comb:abstract.200504gra

Consider a closed curve in the plane that does not intersect itself; by the Jordan Curve Theorem, it bounds a distorted disk. Now consider a closed curve that intersects itself, perhaps several times. Is it the boundary of a distorted disk that overlaps itself? If it is, is that distorted disk essentially unique? The question of when an immersion of the circle can be extended to an immersion of a disk has been studied by several people, notably Titus [C. J. Titus, The combinatorial topology of analytic functions on the boundary of a disk, Acta Math. 106 (1961), 45-64.] and Blank [S. J. Blank, Extending immersions of the circle, Dissertation, Brandeis University, Mass. (1967)]. I will discuss their work and then I will develop combinatorial techniques for answering these questions.