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- Title
INTERSECTIONS OF CURVES ON SURFACES WITH DISK FAMILIES IN HANDLEBODIES.
- Authors
ZABLOW, JOEL
- Abstract
For a surface F bounding a handlebody H, we look at simple closed curves on F which intersect every disk in the handlebody, at least n times (called n-closed curves). We give a finite criterion for a curve to be n-closed. Using this, we derive a sufficiency condition for a Heegaard splitting to be strongly irreducible. We then look at further intersection properties of curves with disk families in H. In particular, we look at the effects of Dehn twists on n-closed curves, and using a finite fixed disk collection $\mathcal{D}\subset H$ as a coordinate system, give heuristics and a counting formula for measuring the number of intersections of the resulting curves, with disks in H. In a certain instance, this yields a partial "grading" on the Dehn twist quandle with respect to the degree of n-closedness.
- Subjects
HANDLEBODIES; DIFFERENTIAL topology; CURVES; DIFFERENTIAL geometry; TOPOLOGY
- Publication
Journal of Knot Theory & Its Ramifications, 2006, Vol 15, Issue 5, p631
- ISSN
0218-2165
- Publication type
Article
- DOI
10.1142/S0218216506004671