We apply the technique of Heegaard Floer Homology to (1,1)-knots (1-bridge knots on the torus) to determine all (1,1)-knots of crossing number up to 12. We also prove miscellaneous results regarding (1,1)-knots, including the existence of a family with trivial Alexander polynomial, and symmetry results.

In addition, we define and study a new class of multi-pointed Heegaard diagrams for links in S^3 that generalizes the classical notion of grid diagrams.

In this thesis, we present some results about $(1,1)$-knots and L-space conjecture. In particular, we prove that (1) L-space twisted torus knots of form $Tp,kp\pm 1l,m$ are closures of $1$-bridge braids; (2) the L-space conjecture holds for the L-spaces obtained from Dehn surgery on closures of iterated $1$-bridge braids, and for $3$-manifolds obtained from Dehn fillings on the hyperbolic $Q$-homology solid torus $v2503$; (3) there are infinitely many $(1,1)$-knots which are topologically slice but not smoothly slice.