Complements on log surfaces

Geodesic Surfaces in Knot Complements

Incompressible Surfaces in Link Complements

A link L in S has a 2n-plat projection for some n, as shown in Figure 1, where a box on the i-th row and j-th column consists of 2 vertical strings with aij left-hand half twist; in other words, it is a rational tangle of slope 1/aij. See for example [BZ]. Let n be the number of boxes in the even rows, so there are n − 1 boxes in the odd rows. Let m be the number of rows in the diagram. It was ...

On Algebraic Hyperbolicity of Log Surfaces

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On 2-Dimensional Homotopy Invariants of Complements of Knotted Surfaces

We prove that ifM is a compact 4-manifold provided with a handle decomposition with 1-skeleton X , and if G is a finite crossed module, then the number of crossed module morphisms from the fundamental crossed module Π2(M,X , ∗) = (π1(X , ∗), π2(M,X , ∗), ∂, ⊲) into G can be re-scaled to a manifold invariant IG (i. e. not dependent on the choice of 1-skeleton), a construction similar to David Ye...

Smooth Surfaces with Non-simply-connected Complements

We give two constructions of surfaces in simply-connected 4-manifolds with non simply-connected complements. One is an iteration of the twisted rim surgery introduced by the first author [7]. We also construct, for any group G satisfying some simple conditions, a simply-connected symplectic manifold containing a symplectic surface whose complement has fundamental group G. In each case, we produ...