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A criterion for satellite 1-genus 1-bridge knots

Authors: Hiroshi Goda, Chuichiro Hayashi and Hyun-Jong Song
Journal: Proc. Amer. Math. Soc. 132 (2004), 3449-3456
MSC (2000): Primary 57M25
Published electronically: April 9, 2004
MathSciNet review: 2073323
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Abstract: Let $K$ be a knot in a closed orientable irreducible 3-manifold $M$. Suppose $M$ admits a genus 1 Heegaard splitting and we denote by $H$ the splitting torus. We say $H$ is a $1$-genus $1$-bridge splitting of $(M,K)$ if $H$intersects $K$ transversely in two points, and divides $(M,K)$ into two pairs of a solid torus and a boundary parallel arc in it. It is known that a $1$-genus $1$-bridge splitting of a satellite knot admits a satellite diagram disjoint from an essential loop on the splitting torus. If $M=S^3$ and the slope of the loop is longitudinal in one of the solid tori, then $K$ is obtained by twisting a component of a $2$-bridge link along the other component. We give a criterion for determining whether a given $1$-genus $1$-bridge splitting of a knot admits a satellite diagram of a given slope or not. As an application, we show there exist counter examples for a conjecture of Ait Nouh and Yasuhara.

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Additional Information

Hiroshi Goda
Affiliation: Department of Mathematics, Tokyo University of Agriculture and Technology, Koganei, Tokyo, 184-8588, Japan

Chuichiro Hayashi
Affiliation: Department of Mathematical and Physical Sciences, Faculty of Science, Japan Women’s University, 2-8-1 Mejiro-dai, Bunkyo-ku, Tokyo, 112-8681, Japan

Hyun-Jong Song
Affiliation: Division of Mathematical Sciences, Pukyong National University, 599-1 Daeyondong, Namgu, Pusan 608-737, Korea

Keywords: $2$-bridge link, twisting operation, $1$-genus $1$-bridge knot, satellite diagram
Received by editor(s): March 17, 2003
Received by editor(s) in revised form: August 11, 2003
Published electronically: April 9, 2004
Additional Notes: This work was supported by Joint Research Project ‘Geometric and Algebraic Aspects of Knot Theory’, under the Japan-Korea Basic Scientific Cooperation Program by KOSEF and JSPS. The authors would like to thank Professor Hitoshi Murakami for giving us this opportunity.
Communicated by: Ronald A. Fintushel
Article copyright: © Copyright 2004 American Mathematical Society