Almost alternating diagrams and fibered links in \(S^3\) (Q2766425)

The authors consider a Seifert surface \(R\) obtained by applying Seifert's algorithm to a connected diagram \(D\) for a link \(L\). They extend results from \textit{D. Gabai} [Comment Math. Helv. 61, 519-555 (1986; Zbl 0621.57003)] to almost alternating links and they give a practical algorithm to determine whether or not a given almost alternating diagram yields a fiber surface via Seifert's algorithm. They show that \(L\) is a fibered link and \(R\) is a fiber surface for \(L\) iff \(R\) is a Hopf plumbing (that is a successive plumbing of a finite number of Hopf bands). First, the authors observe how alternating and almost alternating diagrams behave when the Seifert surfaces obtained by Seifert's algorithm are Murasugi sums. Then they present some examples and show that the results do not extend to 2-almost alternating diagrams. In the appendix they partially answer Adam's open question [Topology Appl. 46, No. 2, 151-165 (1992; Zbl 0766.57003)] and they give an almost alternating diagram for a knot for which it had not been known whether or not it is almost alternating.