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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Closed curves that never extend to proper maps of disks


Author: J. Scott Carter
Journal: Proc. Amer. Math. Soc. 113 (1991), 879-888
MSC: Primary 57M35; Secondary 57N05, 57N10, 57Q35
MathSciNet review: 1070511
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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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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1991-1070511-1
PII: S 0002-9939(1991)1070511-1
Article copyright: © Copyright 1991 American Mathematical Society