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System Analysis of Classification of Prime Knots and Links in Thickened Surfaces of Genus 1 and 2

In this paper, we present a system analysis of approaches to classiﬁcation of prime knots and links in thickened surfaces of genus 1 and 2 obtained by the author in collaboration with S.V. Matveev and V.V. Tarkaev in 2012 – 2020. The algorithm of the classiﬁcation forms structure of the present paper. The results of classiﬁcation are considered within system analysis of the main ideas of key steps of the algorithm. First, we construct prime projections. To this end, we deﬁne a prime link projection, enumerate graphs of special type which embedding in the surface can be a prime projection, enumerate projections in the surface, and show that all obtained projections are prime and not equivalent using some invariants of projections. Second, we construct prime links. To this end, we deﬁne a prime link, construct a preliminary set of diagrams, use invariants of links to form equivalence classes of the obtained diagrams and show that the resulting diagrams are not equivalent, and prove primality of the obtained links. At that, at each step, the used methods and the introduced objects are characterized from viewpoints of two cases (genus 1 and 2), and we distinguish properties that are common for both cases or characteristics of only one of two cases. Note consolidated tables, which systematize the classiﬁed projections with respect to their properties: generative graph, genus, number of components and crossings, existence and absence of bigon that simpliﬁes the further work with the proposed classiﬁcation of projections and links.