Abstract

Until July 2014, all known L-spaces admitted an involution.  Then, through a clever search of the SnapPy census, Dunfield-Hoffman-Licata found examples of asymmetric one-cusped hyperbolic manifolds with two lens space fillings and consequently many asymmetric L-space fillings. Yet since none of these lens space fillings were $S^{3$, so still stood the conjecture that L-space knots in $S}3$ are strongly invertible.

In this talk we present 
(1) a `natural' realization and vast generalization of the Dunfield-Hoffman-Licata examples (joint work with Hoffman and Licata) and
(2) the first construction of asymmetric L-space knots in $S^3$ (joint work with Luecke).

Both of these constructions produce asymmetric one-cusped hyperbolic manifolds with two fillings that are double branched covers of alternating links, though the approaches are different.