Virtual knots and links
DOI link for Virtual knots and links
Virtual knots and links book
This chapter introduces a more combinatorial view where knots and links are defined by crossing information. It deals with a discussion of curves in the plane and gradually build toward a description of oriented virtual links. A more mathematical description of a knot is that a knot is a closed loop in three-dimensional space. The study of knots dates to the 1880s, when Lord Kelvin proposed that atoms were knotted vortices in the ether. Virtual knot and link diagrams are drawn in the plane and represent the possible lists of instructions. This includes diagrams that are projected images of knots and links in three-dimensional space as well as "virtual" knots and links. The chapter explains how to draw virtual link diagrams in a way that avoids ambiguity and describes when several diagrams represent the same set of "instructions". It reviews definition of equivalence relation and equivalence classes.