On L-space knots obtained from unknotting arcs in alternating diagrams

Abstract

Let D be a diagram of an alternating knot with unknotting number one. The branched double cover of S 3 branched over D is an L-space obtained by half integral surgery on a knot KD. We denote the set of all such knots KD by D. We characterize when KD ∈ D is a torus knot, a satellite knot or a hyperbolic knot. In a different direction, we show that for a given n > 0, there are only finitely many L-space knots in D with genus less than n.Definition 1.2. Let (D, c) be an alternating diagram with an unknotting crossing c. Let C be a Conway sphere in D, disjoint from the unknotting arc specified by c. We will call the component of 1 An L-space Y is a rational homology sphere with the simplest possible Heegaard Floer invariant, that is rk HF (Y ) = |H1(Y ; Z)|.