Title: A Criterion for Almost Alternating Links to be Non-splittable

Author:  Tatsuya TSUKAMOTO

Preprint: GWUM-1999-11 

Abstract: The notion of almost alternating links was introduced by C. Adams et al. Here we give a sufficient condition for an almost alternating link diagram to represent a non-splittable link. This solves a question asked in C. Adams, et al, Almost alternating links, Topology Appl. 46 (1992), 151--165. A partial solution for special almost alternating links has been obtained by M. Hirasawa.

As its applications, the Theorem gives us a way to see if a given almost alternating link diagram represents a splittable link without increasing numbers of crossings of diagrams in the process. Moreover, we show that almost alternating links with more than two components are non-trivial. We state them in detail.

To show our theorem, we basically use a technique invented by W. Menasco. However, we also apply ``charge and
discharge method" to our graph-theoretic argument, which is used to prove the four color theorem by K. Appel and W. Haken.