CLASSIFICATION OF PRIME VIRTUAL LINKS OF GENUS 1 WITH AT MOST 4 CLASSICAL CROSSINGS

Abstract

One of the main problems of the knot theory is to classify studied objects, i.e. to construct a table of all inequivalent objects taking into account parameters represented some properties, as well as a list of invariants of the tabulated objects. The goal of this paper is to classify all genus 1 prime virtual links having virtual link diagrams with at most 4 classical crossings. The problem of classification is difficult, because there is no universal method to decide if two given objects are equivalent or not. We generalise Kauffman bracket of virtual link diagrams in order to obtain an invariant, which  is enough to prove that constructed table contains only inequivalent objects. To this end, we propose an algorithm to compute the numbers of trivial and nontrivial curves. The results of the paper can be introduced into research on the proteins by means of a method to represent proteins as virtual links.