New presentations of a link and virtual link

Abstract

An embedding presentation of a diagram is introduced, which has proved to be a unique presentation of a diagram. Let [Formula: see text] be a set of all diagrams, called also links in this paper. An algebraic system [Formula: see text] is constructed. In fact, a link in [Formula: see text] (or [Formula: see text]) is the equivalent class [Formula: see text] where [Formula: see text] is one of its embedding presentations. Based on [Formula: see text], Reduction Crossing Algorithm is proposed which is used to reduce the number of crossings in an embedding presentation by introducing a main tool called a pass replacement. For an infinite set of unknots [Formula: see text], each [Formula: see text] in [Formula: see text] can be transformed into the trivial unknot in at most [Formula: see text] by applying the algorithm where [Formula: see text] is a constant, [Formula: see text] and [Formula: see text]. As special consequences, three unknots are unknotted, which are Goeritz’s unknot, Thistlethwaite’s unknot and Haken’s unknot (image courtesy of Cameron Gordon). Moreover, an infinite family of unknots [Formula: see text] are unknotted in [Formula: see text] time. In addition, unique presentations of a virtual link, an oriented link and oriented virtual link are introduced, respectively.