A Murasugi decomposition for achiral alternating links

Abstract

We prove that an achiral alternating link can be decomposed in a strong sense as a Murasugi sum of a link and its mirror image. The proof relies on our theory of Murasugi atoms. We introduce a notion of a bond between atoms, called adjacency. This relation is expressed by a graph, the adjacency graph, which is an isotopy invariant. A well-defined link type, called a molecule, is associated to any connected subgraph of the adjacency graph. The Flyping Theorem of Menasco and Thistlethwaite is the main tool used to prove the isotopy invariance of atoms, molecules and the adjacency graph. The action of flypes on the adjacency graph and the invariance of the collection of molecules under flypes are the main ingredients of the proof of the decomposition theorem.