For any integer N ^ 3 we will construct a 3-manifold which can be described as +1 surgery on N distinct knots in S. We will also give examples of 3-manifolds which are i^-fold cyclic branched covers of S over 2 distinct knots. Brakes (2) discovered the first examples of 3-manifolds with multiple knot surgery descriptions. Our construction is much different and follows directly from the construc...

This paper and its prequel (“Part I”) prove a generalization of the usual gluing theorem for two index 1 pseudoholomorphic curves U+ and U− in the symplectization of a contact 3-manifold. We assume that for each embedded Reeb orbit γ, the total multiplicity of the negative ends of U+ at covers of γ agrees with the total multiplicity of the positive ends of U− at covers of γ. However, unlike in ...

We compute the nonabelian SL2(C)-character varieties of rational knots C(2n+1,2m,2) in Conway notation, where m and n are non-zero integers. By using real points on these varieties, we study left orderability fundamental groups cyclic branched covers C(2n+1,2m,2).

We show that under suitable conditions a branched cover satisses the same upper curvature bounds as its base space. First we do this when the base space is a metric space satisfying Alexandrov's curvature condition CAT() and the branch locus is complete and convex. Then we treat branched covers of a Riemannian manifold over suitable mutually orthogonal submanifolds. In neither setting do we req...

We show that over an algebraically closed field of characteristic not equal to 2, homological projective duality for smooth quadric hypersurfaces and double covers spaces branched is a combination two operations: one interchanges hypersurface with its classical dual the other cover along it.

It is known that if any prime power branched cyclic cover of a knot in S is a homology sphere, then the knot has vanishing Casson-Gordon invariants. We construct infinitely many examples of (topologically) non-slice knots in S whose prime power branched cyclic covers are homology spheres. We show that these knots generate an infinite rank subgroup of F(1.0)/F(1.5) for which Casson-Gordon invari...

We show that quasi-alternating links arise naturally when considering surgery on a strongly invertible L-space knot (that is, a knot that yields an L-space for some Dehn surgery). In particular, we show that for many known classes of L-space knots, every sufficiently large surgery may be realized as the two-fold branched cover of a quasi-alternating link. Consequently, there is considerable ove...

Let B be a smooth, connected, projective complex curve of genus h. For w ≥ 2d we prove the irreducibility of the Hurwitz stack H Sd (B) parametrizing degree d covers of B simply-branched over w points, and with monodromy group Sd.