Quasoids in Knot Theory

Abstract

This paper is devoted to the definition and construction of quasoids, which are algebraic objects generating invariants of oriented knots and links. Such an invariant can be expressed in terms of the number of proper colorings of the regions into which a knot diagram divides the 2-sphere. A coloring with elements of a set X is proper if the colors of all four regions in a neighborhood of each crossing point of the diagram are matched by means of a function Q: X×X×X → X. This function is called a quasoid over the set X. In this paper,we construct two infinite series of quasoids. The first series is formed by linear quasoids over finite rings. The second series consists of quasoids generated by finite biquasiles. The invariants of knots and links generated by quasoids are nontrivial and can be used to distinguish knots. We show that all knots and links admitting diagrams with at most six crossings are distinguished by linear quasoids over ℤⁿ, where n ≤ 11. We give results of the computer enumeration of all different quasoids over sets whose cardinality does not exceed 4.