High distance bridge surfaces

Abstract

Abstract. Given integers b, c, g, and n, we construct a manifold M containing a c-component link L so that there is a bridge surface Σ for (M, L) of genus g that intersects L in 2b points and has distance at least n. More generally, given two possibly disconnected surfaces S and S ′ , each with some even number (possibly zero) of marked points, and integers b, c, g, and n, we construct a compact, orientable manifold M with boundary S ∪ S ′ such that M contains a c-component tangle T with a bridge surface Σ of genus g that separates ∂M into S and S ′ , |T ∩ Σ| = 2b and T intersects S and S ′ exactly in their marked points, and Σ has distance at least n.

Width is not additive

“…By incompressibility of F K and irreducibility of N K ,ˇbounds a disk in F K . Hence, we have conclusion (3). By a similar argument, if˛has endpoints b C 1 and c C s or b 1 and c s we have conclusion (4).…”

Hyperbolic manifolds containing high topological index surfaces

“…The existence of such a block will follow from a result of Blair, Tomova, and Yoshizawa. It is a special case of Corollary 5.3 from [9]. Define the bridge disk set of V (resp., W ), denoted BD V ⊂ AC(B) (resp., BD W ), as the set of all vertices either corresponding to essential simple closed curves in B ′ that bound embedded disks in V L (resp., W L), or corresponding to bridge arcs in B ′ .…”

Knots with compressible thin levels

“…Note that after a single compression S 3 is isotopic to S 1 ∪ S 2 . The thin-thick tuple for k ′ is (10,8,10,4,6), which gives w(k ′ ) = 78 by equation (1). Let k denote any other embedding of K. We will use a case-by-case analysis to show w(k) ≥ 78.…”