Mathematical Proceedings of the Cambridge Philosophical Society

Dehn surgeries on knots which yield lens spaces and genera of knots

a1 The Graduate School of Science and Technology, Kobe University, 1-1 Rokkodai, Nada, Kobe 657-8501, Japan; e-mail:
a2 Department of Mathematics and Mathematics Education, Hiroshima University, 1-1-1 Kagamiyama, Higashi-hiroshima 739-8524, Japan; e-mail:


It is an interesting open question when Dehn surgery on a knot in the 3-sphere S3 can produce a lens space (see [10, 12]). Some studies have been made for special knots; in particular, the question is completely solved for torus knots [21] and satellite knots [3, 29, 31]. It is known that there are many examples of hyperbolic knots which admit Dehn surgeries yielding lens spaces. For example, Fintushel and Stern [8] have shown that 18- and 19-surgeries on the (−2, 3, 7)-pretzel knot give lens spaces L(18, 5) and L(19, 7), respectively. However, there seems to be no essential progress on hyperbolic knots. It might be a reason that some famous classes of hyperbolic knots, such as 2-bridge knots [26], alternating knots [5], admit no surgery yielding lens spaces.

In this paper we focus on the genera of knots to treat the present condition methodically and show that there is a constraint on the order of the fundamental group of the resulting lens space obtained by Dehn surgery on a hyperbolic knot. Also, this new standpoint enables us to present a conjecture concerning such a constraint, which holds for all known examples.

(Received June 24 1999)