Abstract
In this article, we present the result of the first step of tabulation of prime links in the thickened surface of genus 2 that admit diagrams with no more than 4 crossings. Namely, we describe all three steps of tabulation of prime link projections in the surface of genus 2 with no more than 4 crossings. First, we define primality of a link projection in the surface of genus 2. Second, we tabulate prime link projections in the surface of genus 2 with no more than 4 crossings. For this purpose, it is sufficient to consider graphs having special type and enumerate all possible embeddings of the graphs into the surface of genus 2 giving prime link projections. At this step, we prove some auxiliary statements to simplify enumeration of the embeddings. Finally, we show that all obtained projections are nonequivalent in the sense of homeomorphism of the surface of genus 2 onto itself. Our main result states that there exist exactly 15 pairwise nonequivalent prime link projections in the surface of genus 2 with no more than 4 crossings. Several new and known tricks allow rigorously theoretically prove the completeness of the obtained tabulation, as well as to keep the process within reasonable limits. Further, we intend to use the obtained table to classify prime diagrams, i.e. to obtain table of prime links.
Highlights
In the knot theory, one of the oldest and the most important problems is to recognize a knot, i. e., to associate the considered object with a unique tabulated one
In contrast to the case of knots and links in the 3-dimensional sphere, there is a gap between global knots and links in the sense of tabulation
We propose to tabulate virtual knots taking into account both numerical characteristics, i. e. the number of classical crossings as usual, and the genus of a knot, see the articles [9, 10] for classifications of prime knots in the thickened torus and the thickened surface of genus 2, respectively
Summary
One of the oldest and the most important problems is to recognize a knot (or a link), i. e., to associate the considered object with a unique tabulated one. Most of the obtained classifications consider knots and links in the 3-dimensional sphere, see [1,2,3]. The works [7] and [8] present perfect classifications of virtual knots ordered taking into account the number of classical crossings and obtain a list of some characteristics of each knot. We begin tabulation of prime links in the thickened surface of genus 2 For this purpose, in this article, we present the result of the first step, i. E. we obtain a classification of prime projections of links in the surface of genus 2 with no more than 4 crossings. The complement T2 \ C is formed by two copies of a torus To with a hole
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