Mathematical Proceedings of the Cambridge Philosophical Society

Genus one 1-bridge positions for the trivial knot and cabled knots

a1 Department of Mathematics, Faculty of Science, Gakushuin University, Mejiro 1-5-1, Toshima-ku, Tokyo 171, Japan; e-mail:


Let (Vi, ti) be a pair consisting of a solid torus Vi and an unknotted arc ti properly imbedded in Vi for i=1 and 2. Taking the union of these pairs we obtain a pair (M, K) of the 3-dimensional sphere or a lens space M and a knot K. We call such a knot K a genus one 1-bridge knot. In this paper we determine when a genus one 1-bridge knot is the trivial knot or a cabled knot.

(Received February 20 1997)
(Revised July 29 1997)


1 This research was partially supported by Grant-in-Aid for Scientific Research (No. 08740069), Ministry of Education, Science and Culture.