Hyperbolic H-knots in non-trivial lens spaces are not determined by their complement

Abstract

If a knot K in a lens space M is not determined by its complement then there exists a non-trivial r-Dehn surgery on K which produces M. If this surgery conserves a Heegaard diagram of M, i.e. there exists a Heegaard solid torus V which is still a Heegaard solid torus after the r-Dehn surgery, the knot is said to be a H-knot. The knot of S. A. Bleiler, C. D. Hodgson and J. R. Weeks [2] is such a knot, and is hyperbolic. In [12] it is shown that non-hyperbolic knots are determined by their complements in lens spaces, except axes in L(p,q) when q2≢±1modp. Here, the goal is to see that hyperbolic H-knots are not determined by their complements, i.e. there is no automorphism onto the complement which sends the meridian slope to the r-slope.