A Criterion for Almost Alternating Links to Be Non-splittable

The notion of almost alternating links was introduced by C. Adams et al ([1]). Here we give a sufficient condition for an almost alternating link diagram to represent a non-splittable link. This solves a question asked in [1]. A partial solution for special almost alternating links has been obtained by M. Hirasawa ([4]). As its applications, Theorem 2.3 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 nontrivial. In Section 2, we state them in detail. To show our theorem, we basically use a technique invented by W. Menasco (see [5, 6]). We review it in Section 3. However, we also apply “ charge and discharge method” to our graph-theoretic argument, which is used to prove the four color theorem in [3].