Classification of Prime Projections of Knots in the Thickened Torus of Genus 2 with at most 4 Crossings

Abstract

We begin classification of prime knots in the thickened torus of genus 2 having diagrams with at most 4 crossings. To this end, it is enough to construct a table of prime knot projections with at most 4 crossings, and use the table to obtain table of prime diagrams, i. e. table of prime knots. In this paper, we present the result of the first step, i. e. we construct a table of prime projections of knots in the thickened torus of genus 2 having at most 4 crossings. First, we introduce definition of prime projection of a knot in the thickened torus of genus 2. Second, we construct a table of prime projections of knots in the thickened torus of genus 2 having at most 4 crossings. To this end, we enumerate graphs of special type and consider all possible embeddings of the graphs into the torus of genus 2 that lead to prime projections. In order to simplify enumeration of the embeddings, we prove some auxiliary statements. Finally, we prove that all obtained projections are inequivalent. Several known and new tricks allow us to keep the process within reasonable limits and rigorously theoretically prove the completeness of the constructed table