We study the fibration of augmented link complements. Given the diagram of an augmented link we associate a spanning surface and a graph. We then show that this surface is a fiber for the link complement if and only if the associated graph is a tree. We further show that fibration is preserved under Dehn filling on certain components of these links. This last result is then used to prove that within a very large class of links, called locally alternating augmented links, every link is fibered. arXiv:1109.3084v2 [math.GT]

We prove that if the fundamental group of an orientable finite volume hyperbolic 3-manifold has finite index in the reflection group of a right-angled ideal polyhedron in H 3 then it has a co-final tower of finite sheeted covers with positive rank gradient. The manifolds we consider are also known to have co-final towers of covers with zero rank gradient.

We show a combinatorial argument in the diagram of large class of links, including satellite and hyperbolic links, where for each of which the tunnel number is the minimum possible, the number of its components minus one.

In this paper we define alternating Kauffman states of links and we characterize when the induced state surface is a fiber. In addition, we give a different proof of a similar theorem of Futer, Kalfagianni and Purcell on homogeneous states.2010 Mathematics Subject Classification. 57M25, 57M15, 57M50.

We prove that the rank of the fundamental group of an augmented link complement equals its Heegaard genus. This is achieved by showing that the Heegaard genus equals the number of link components. A straightforward consequence is that these manifolds satisfy a conjecture of Shalen.