Virtual Knot Groups and Combinatorial Knots

Abstract

Kauffman [16] and Kim [17] defined the group of a virtual knot by extending, in a natural way, the Wirtinger presentation of the fundamental group of classical knot. In this paper we present the group of a virtual knot by using the concept of combinatorial knot, introduced by Toro [21]. We show the advantages of this approach, that provides natural algorithms. We present examples of combinatorial knots whose groups have properties that are false, or unknown, in the category of the classical knots.

Keywords: Knots, Virtual Knots, Combinatorial Knots, Knot Groups, Virtual Knot Groups.