✦ LIBER ✦

Rotation numbers for curves on a torus

✍ Scribed by Hitoe Tanio; Osamu Kobayashi

Book ID: 104642804
Publisher: Springer
Year: 1996
Tongue: English
Weight: 454 KB
Volume: 61
Category: Article
ISSN: 0046-5755
DOI: 10.1007/bf00149414

No coin nor oath required. For personal study only.

✦ Synopsis

We define a regular homotopy invariant of closed curves on a surface, and give a formula for the rotation number of closed curves on toms, which is analogous to the Whitney formula for planar curves. As an application, we show a necessary condition for a Gauss word to be realized on toms.

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We show that if \(G\) is a graph embedded on the torus \(S\) and each nonnullhomotopic closed curve on \(S\) intersects \(G\) at least \(r\) times, then \(G\) contains at least \(\left\lfloor\frac{3}{4} r\right\rfloor\) pairwise disjoint nonnullhomotopic circuits. The factor \(\frac{3}{4}\) is best