A Murasugi decomposition for achiral alternating links

Cam Van Quach Hongler and Claude Weber

Vol. 222 (2005), No. 2, 317–336

DOI: 10.2140/pjm.2005.222.317

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.

Keywords

alternating links, achiral links, invertible links, Murasugi decomposition, flype conjecture

Mathematical Subject Classification 2000

Primary: 57M25

Milestones

Received: 10 January 2004

Revised: 13 January 2005

Accepted: 5 February 2005

Published: 1 December 2005