On (1,1)-knots and L-space conjecture

Abstract

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±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.