Hostname: page-component-cd9895bd7-jn8rn Total loading time: 0 Render date: 2024-12-25T08:15:37.586Z Has data issue: false hasContentIssue false
  • English
  • Français

ON 2-ADJACENCY RELATION OF TWO-BRIDGE KNOTS AND LINKS

Part of: Low-dimensional topology

Published online by Cambridge University Press:  01 February 2008

ICHIRO TORISU
ICHIRO TORISU*
Affiliation:
Naruto University of Education, 748, Nakajima, Takashima, Naruto-cho, Naruto-shi, 772-8502, Japan (email: [email protected])
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We give a necessary condition for a two-bridge knot or link S(p,q) to be 2-adjacent to another two-bridge knot or link S(r,s). In particular, we show that if the trivial knot or link is 2-adjacent to S(p,q), then S(p,q) is trivial, that if S(p,q) is 2-adjacent to its mirror image, then S(p,q) is amphicheiral, and that for a prime integer p, if S(p,q) is 2-adjacent to S(r,s), then S(p,q)=S(r,s) or S(r,s)=S(1,0).

Keywords

n-adjacenttwo-bridge knotlinkDehn surgery

MSC classification

Secondary: 57M25: Knots and links in $S^3$
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 84 , Issue 1 , February 2008 , pp. 139 - 144
Copyright
Copyright © 2008 Australian Mathematical Society

References

[1]Culler, M., Gordon, C., Luecke, J. and Shalen, P., ‘Dehn surgery on knots’, Ann. of Math. (2) 125 (1987), 237–300.CrossRefGoogle Scholar
[2]Howards, H. and Luecke, J., ‘Strongly n-trivial knots’, Bull. London Math. Soc. 34 (2002), 431–437.CrossRefGoogle Scholar
[3]Kalfagianni, E., ‘Crossing changes of fibered knots’, Preprint.Google Scholar
[4]Kalfagianni, E. and Lin, X.-S., ‘Knot adjacency and satellites’, Topology Appl. 138 (2004), 207–217.CrossRefGoogle Scholar
[5]Kalfagianni, E. and Lin, X.-S., ‘Knot adjacency, genus and essential tori’, Pacific J. Math. 228 (2006), 251–275.CrossRefGoogle Scholar
[6]Montesinos, J. M., ‘Surgery on links and double branched coverings of S 3’, Ann. of Math. Stud. 84 (1975), 227–259.Google Scholar
[7]Torisu, I., ‘The determination of the pairs of two-bridge knots or links with Gordian distance one’, Proc. Amer. Math. Soc. 126 (1998), 1565–1571.CrossRefGoogle Scholar
[8]Torisu, I., ‘On nugatory crossings for knots’, Topology Appl. 92 (1999), 119–129.CrossRefGoogle Scholar
[9]Torisu, I., ‘On strongly n-trivial 2-bridge knots’, Math. Proc. Cambridge Philos. Soc. 137 (2004), 613–616.CrossRefGoogle Scholar
[10]Torisu, I., ‘A note on strongly n-trivial links’, J. Knot Theory Ramifications 14 (2005), 565–569.CrossRefGoogle Scholar
[11]Torisu, I., ‘Two-bridge links with strong triviality’, Tokyo J. Math. 29 (2006), 233–237.CrossRefGoogle Scholar
[12]Tsutsumi, Y., ‘Strongly n-trivial links are boundary links’, Tokyo J. Math. 30 (2007), 343–350.CrossRefGoogle Scholar