A criterion for almost alternating links to be non-splittable

Abstract

The notion of almost alternating links was introduced by C. Adams et al ([1]). We give a sufficient condition for an almost alternating link diagram to represent a non-splittable link. This solves a question asked in [1, 3]. A partial solution for special almost alternating links has been obtained by M. Hirasawa ([5]). As an application, Theorem 2.2 gives us a simple finite algorithm to decide if a given almost alternating link diagram represents a splittable link without increasing the number of crossings of diagrams in the process. Moreover, we show that almost alternating links with more than two components are non-trivial. In Section 2, we state these results in detail. To prove our theorem, we use a technique invented by W. Menasco (see [6, 7]), this is reviewed in Section 3. We also apply “the charge and discharge method” to our graph-theoretic argument, which was used to prove the four color theorem in [4], we explain this in Section 4.