Problem of knot complement in lens spaces

Daniel Matignon

Knots in Washington XXVII; 3rd Japan-USA Workshop in Knot Theory
January 9-11, 2009
George Washington University
Washington, DC, USA

Organizers
Yoshiyuki Ohyama (TWCU), Jozef H. Przytycki (GWU), Yongwu Rong (GWU), Alexander Shumakovitch (GWU), Kouki Taniyama (Waseda and GWU), Tatsuya Tsukamoto (Osaka IT), Hao Wu (GWU), Akira Yasuhara (Tokyo GU)

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Problem of knot complement in lens spaces
by
Daniel Matignon
Université de Marseille-Provence

A knot k in a 3-manifold M is not determined by its complement if there exists a homeomorphism between its complement and the complement of another knot k', but no homeomorphism from (M, k) to (M, k'). The problem of the existence of such a knot was initially settled by Tietze in 1908 for M=S³, then solved by Gordon and Luecke in 1989; and later the problem was generalized for all compact and orientable 3-manifolds. The knot complement problem is closely relative to the existence of cosmetic pairs, i.e. a non-trivial Dehn surgery on k yields a 3-manifold homeomorphic to M.

This paper concerns the study of the cosmetic pairs when the M's are lens spaces. We consider separately the knots according to their geometric type : Seifert fibered knots, satellite knots or hyperbolic knots. For the two former cases, we explicitly give exhaustive and infinite families of such cosmetic pairs. Each pair in these families is a counter-example to the Gordon's conjecture, which asserts that the homeomorphims between M and M_k(r) are orientation reversing. In the latter case, We show that there is no cosmetic pair (M, k) when M is a lens space and k is a fibered knot.

Date received: November 11, 2008