Closed curves that never extend to proper maps of disks

Carter, J. Scott

doi:10.1090/S0002-9939-1991-1070511-1

Closed curves that never extend to proper maps of disks
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Proc. Amer. Math. Soc. 113 (1991), 879-888 Request permission

Abstract:

If a closed curve in an orientable surface bounds a disk in a handlebody, then the double points on the boundary admit certain pairings that are called filamentations. Intersection numbers are associated to the filamentations; these numbers provide a necessary criterion for the existence of a disk bounded by a given curve. As an application, a closed curve with three generic double points in a surface of genus 2 is given that bounds no disk in a handlebody. This is the most simple example of a closed curve that does not bound a disk. The example is generalized to find for each $n > 3$, a closed curve with $n$ crossings that does not bound a disk.

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