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A criterion for satellite 1-genus 1-bridge knots |
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| Authors: | Hiroshi Goda Chuichiro Hayashi Hyun-Jong Song |
| Institution: | Department of Mathematics, Tokyo University of Agriculture and Technology, Koganei, Tokyo, 184-8588, Japan ; Department of Mathematical and Physical Sciences, Faculty of Science, Japan Women's University, 2-8-1 Mejiro-dai, Bunkyo-ku, Tokyo, 112-8681, Japan ; Division of Mathematical Sciences, Pukyong National University, 599-1 Daeyondong, Namgu, Pusan 608-737, Korea |
| Abstract: | Let  be a knot in a closed orientable irreducible 3-manifold  . Suppose  admits a genus 1 Heegaard splitting and we denote by  the splitting torus. We say  is a  -genus  -bridge splitting of  if  intersects  transversely in two points, and divides  into two pairs of a solid torus and a boundary parallel arc in it. It is known that a  -genus  -bridge splitting of a satellite knot admits a satellite diagram disjoint from an essential loop on the splitting torus. If  and the slope of the loop is longitudinal in one of the solid tori, then  is obtained by twisting a component of a  -bridge link along the other component. We give a criterion for determining whether a given  -genus  -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. |
| Keywords: | $2$-bridge link twisting operation $1$-genus $1$-bridge knot satellite diagram |
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