Epimorphisms between 2–bridge knot groups and their crossing numbers

Suzuki, Masaaki¹

¹Department of Frontier Media Science, School of Interdisciplinary Mathematical Sciences, Meiji University, 4-21-1 Nakano, Tokyo 164-8525, Japan

Algebraic and Geometric Topology, Tome 17 (2017) no. 4, pp. 2413-2428

Résumé

Suppose that there exists an epimorphism from the knot group of a 2–bridge knot  K onto that of another knot K′. We study the relationship between their crossing numbers c(K) and c(K′). More specifically, it is shown that c(K) is greater than or equal to 3c(K′), and we estimate how many knot groups a 2–bridge knot group maps onto. Moreover, we formulate the generating function which determines the number of 2–bridge knot groups admitting epimorphisms onto the knot group of a given 2–bridge knot.

Détail

DOI : 10.2140/agt.2017.17.2413

Classification : 57M25, 57M27
Keywords: epimorphism, $2$–bridge knot, knot group, crossing number

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