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Two maps with large representativity on one surface

✍ Scribed by R. Bruce Richter; Gelasio Salazar

Publisher
John Wiley and Sons
Year
2005
Tongue
English
Weight
120 KB
Volume
50
Category
Article
ISSN
0364-9024

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✦ Synopsis

Abstract

We show that, for each orientable surface Ξ£, there is a constant c~Ξ£~ so that, if G~1~ and G~2~ are embedded simultaneously in Ξ£, with representativities r~1~ and r~2~, respectively, then the minimum number cr(G~1~, G~2~) of crossings between the two maps satisfies

This refines earlier estimates by Negami. Furthermore, we provide a counterexample to a conjecture of Archdeacon and Bonnington by exhibiting, for each k, embeddings G~1~ and G~2~ in the double torus so that, if we force all the vertices of G~1~ to be in the same face of G~2~, then the number of crossings between G~1~ and G~2~ is at least k · cr(G~1~, G~2~). Β© 2005 Wiley Periodicals, Inc.

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