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 $T_{p,kp\pm 1}^{l,m}$ 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 $\mathbf{Q}$-homology solid torus $v2503$; (3) there are infinitely many $(1,1)$-knots which are topologically slice but not smoothly slice.