A criterion for satellite 1-genus 1-bridge knots
HTML articles powered by AMS MathViewer
- by Hiroshi Goda, Chuichiro Hayashi and Hyun-Jong Song PDF
- Proc. Amer. Math. Soc. 132 (2004), 3449-3456 Request permission
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.References
- Mohamed Ait Nouh and Akira Yasuhara, Torus knots that cannot be untied by twisting, Rev. Mat. Complut. 14 (2001), no. 2, 423–437. MR 1871306, DOI 10.5209/rev_{r}ema.2001.v14.n2.16993
- D. B. A. Epstein, Curves on $2$-manifolds and isotopies, Acta Math. 115 (1966), 83–107. MR 214087, DOI 10.1007/BF02392203
- H. Goda, C. Hayashi and H. Song, Dehn surgeries on $2$-bridge links which yield reducible $3$-manifolds, preprint.
- Chuichiro Hayashi, Genus one $1$-bridge positions for the trivial knot and cabled knots, Math. Proc. Cambridge Philos. Soc. 125 (1999), no. 1, 53–65. MR 1645517, DOI 10.1017/S0305004198002916
- Chuichiro Hayashi, Satellite knots in 1-genus 1-bridge positions, Osaka J. Math. 36 (1999), no. 3, 711–729. MR 1740829
- C. Hayashi, $1$-genus $1$-bridge splittings for knots, to appear in Osaka J. Math. 41 (2004).
- Kanji Morimoto, On minimum genus Heegaard splittings of some orientable closed $3$-manifolds, Tokyo J. Math. 12 (1989), no. 2, 321–355. MR 1030498, DOI 10.3836/tjm/1270133184
- Kanji Morimoto and Makoto Sakuma, On unknotting tunnels for knots, Math. Ann. 289 (1991), no. 1, 143–167. MR 1087243, DOI 10.1007/BF01446565
Additional Information
- Hiroshi Goda
- Affiliation: Department of Mathematics, Tokyo University of Agriculture and Technology, Koganei, Tokyo, 184-8588, Japan
- Email: goda@cc.tuat.ac.jp
- 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
- Email: hayashic@fc.jwu.ac.jp
- Hyun-Jong Song
- Affiliation: Division of Mathematical Sciences, Pukyong National University, 599-1 Daeyondong, Namgu, Pusan 608-737, Korea
- Email: hjsong@pknu.ac.kr
- 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
- © Copyright 2004 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 132 (2004), 3449-3456
- MSC (2000): Primary 57M25
- DOI: https://doi.org/10.1090/S0002-9939-04-07505-7
- MathSciNet review: 2073323