A question on (1,1) bridge Knot 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.
 A: All two-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.
A: 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 Baker's work is a great place to find examples of this. Specifically, in this paper

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*Kenneth L. Baker, Surgery descriptions and volumes of Berge knots II: Descriptions on the minimally twisted five chain link,  Journal of Knot Theory and Its Ramifications 17 No. 09 (2008) pp. 1099–1120, doi:10.1142/S021821650800652X, arXiv:math/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.
