Construction of spines of two-bridge link complements and upper bounds of their Matveev complexities
The notion of complexity was introduced by S. Matveev in 1990. This is now known as the Matveev complexity, and it is a very important invariant for 3-manifolds in geometric topology. But determining the Matveev complexity of a given 3-manifold is quite difficult in general. Thus, providing sharp bounds for the Matveev complexities of 3-manifolds is an important problem. In fact, many authors worked on this. In particular, Petronio and Vesnin gave a certain lower bound for the Matveev complexity of a two-bridge link complement.NEWLINENEWLINEIn this article the authors construct an almost-simple spine of a two-bridge link complement. This naturally gives upper bounds for the Matveev complexities of two-bridge link complements. Then using their construction together with the previous known lower bounds for the Matveev complexity of two-bridge link complements, they are able to determine the complexities for an infinite sequence of two-bridge links corresponding to the continued fractions of the form \([2,1,\dots,1,2]\). They also give upper bounds for the 3-manifolds obtained as meridian-cyclic branched coverings of the 3-sphere along two-bridge links.
scientific article; zbMATH DE number 6643502