A characterization of alternating link exteriors in terms of cubed complexes

Abstract

We give a characterization of alternating link exteriors in terms of cubed complexes. To this end, we introduce the concept of a "signed BW cubed-complex", and give a characterization for a signed BW cubed-complex to have the underlying space which is homeomorphic to an alternating link exterior. IntroductionRecently, Greene [5] and Howie [8], independently, established intrinsic characterizations of alternating links in terms of a pair of spanning surfaces, answering an old question of R. H. Fox. These results can be regarded as characterizations of alternating link exteriors which have marked meridians (see [8, Theorem 3.2]).The purpose of this paper is to give a characterization of alternating link exteriors from the viewpoint of cubed complexes. Our starting point is a cubical decomposition of alternating link exteriors, which is originally due to Aitchison, and is used by Agol [2], Adams [1], Thurston [10], Yokota [11,12] and Sakuma-Yokota [9]. Thus we call it the Aitchison complex. The Aitchison complex for an alternating link is actually a mapping cylinder of the natural map from the boundary of the exterior of the alternating link onto the Dehn complex. For a detailed description and historical background, see [9].In this paper, we introduce the concepts of a signed BW squared-complex (or an SBW squared-complex, for short) and a signed BW cubed-complex (or an SBW cubed-complex, for short), and give a combinatorial description of the Dehn complex and the Aitchison complex as an SBW squared-complex and an SBW cubedcomplex, respectively. The main theorem gives a necessary and sufficient condition for a given SBW cubed-complex to be isomorphic to the Aitchison complex of some alternating link exterior (Theorem 4.1). This implies a characterization of alternating link exteriors in terms of cubed complexes (Corollary 4.2). This paper is organized as follows. In Section 2, we give an intuitive description of the Aitchison complex and the Dehn complex following [10,11]. In Section 3, we introduce the SBW squared-complex and the SBW cubed-complex, and describe the Dehn complex and the Aitchison complex in terms of the SBW squared-complex and the SBW cubed-complex, respectively. In Section 4, we prove the main theorem.The author would like to thank his supervisor, Makoto Sakuma, for valuable suggestions. He would also like to thank Naoki Sakata and Takuya Katayama for their support and encouragement.2. An intuitive description of the Aitchison complexes arXiv:1804.03473v1 [math.GT]