On the fibration of augmented link complements

Abstract

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]

Fiber detection for state surfaces

“…This inductive approach is very similar in spirit to the methods used by Ozawa [13]. It is also fruitfully exploited in a recent preprint of Girão to prove a fibering criterion for augmented links [10].…”

Fiber surfaces from alternating states

“…In section 3 we give a different, homological proof, of the following theorem of Futer, Kalfagianni and Purcell [2] on homogeneous states. The techniques we use in our proof are similar to the ones in the paper [4] by the first author, where he studies the fibration of augmented link complements.…”

“…In [1] a much simpler proof is given: it is proved inductively via Murasugi sums together with Theorem 3 to deduce fibering information. Some of these ideas were also independently used in the work of the first author [4] and in the previous section. The proof we present is a consequence of Stallings' fibration criteria [8].…”

“…Later, Futer [1] gave a different, much simpler proof of this result. Based on the work of Gabai [3] and Stallings [8], the first author [4] studied fibration of state surfaces of augmented links. This paper is concerned with the study of another class of state surfaces, which we now describe.…”

Heegaard genus and rank of augmented link complements

“…Cheesebro-DeBlois-Wilton [3] proved that hyperbolic augmented links satisfy the virtual fibering conjecture (recently proved in full generality by Agol [1]). More recently, the author has given a criterion to determine when the complements of these links are fibered [6]. We describe these links now.…”