Heegaard genus and rank of augmented link complements

Abstract

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.

On links minimizing the tunnel number

“…As far as we know, the conjecture remains open for link exteriors in S 3 . The first author [2] proved this conjecture to be true for augmented links. Theorem 1.1 shows that this is also the case for band links, as stated in the following corollary.…”

“…Although this method may not find optimal bounds in general, it seems to be useful in other classes of links. For example, in [2] the tunnel number of augmented links has been determined. The method of percolation can, after appropriate choice of vertices, be used to obtain the the results therein.…”