Question

Hi, everyone. I am interested in the complement of (1,1) bridge knot in a lens space, $S^{3}$. Is there one (1,1) bridge knot in $S^{3}$ or lens space such that its complement is hyperbolic?

Note: A knot $K$ in $S^{3}$ or Lens space is (1,1) if for the standard genus 1 Heegaard splitting of $S^{3}$ or lens space, $K$ intersects each solid torus only one arc which is boundary parallel.

Answer 1

All 2-bridge knots in $S^3$ are $(1,1)$-knots in $S^3$. This is assigned as an exercise here. All two-bridge knots, other than the $(2,2k+1)$-torus knots, are hyperbolic.

Answer 2

Actually, there are a lot of these kinds of knots.

First, let's set some notation. Start with two manifolds $M_1$ and $M_2$ related by a Dehn surgery along an embedded curve $K$ in $M_1$. After removing a neighborhood of K, we glue in a solid torus $T$ along $\partial N(K)$ to obtain $M_2$. We say the core $K'$ of $T$ is the dual knot to $K$ in $M_2$. The dual knots to many Berge knots are (1,1) knots lens space. Ken Bakers work is a great place to find examples of this. Specifically, in this paper http://front.math.ucdavis.edu/math.GT/0509055 all knots in families III-VI and VIII-XII are hyperbolic knots in $S^3$ that are dual to (1,1) knots in Lens spaces. Forcing the dual knots to be hyperbolic as well.

If you want a more concrete example. The (-2,3,7) pretzel knot is hyperbolic and admits 2 lens space surgeries. (This was first observed by Fintushel and Stern.) The lens spaces are (18,7) and (19,7) and the dual knots to the (-2,3,7) in these lens spaces are (1,1) knots.