NORMALIZATION AND EXPANSION OF KAUFMAN POLYNOMIALSTO DISORIENTED LINKS AND KNOTS

Abstract

This article introduces a regular isotopy invariant for disoriented links, which is a significant contribution to the development of disoriented knot theory. Disoriented diagrams arise in classical and virtual knot theory when calculating Jones and HOMFLY polynomials using state summation for the link diagrams. The article defines a two-variable polynomial invariant, MK, and a normalized polynomial ambient isotopy invariant, NK, for disoriented link diagram K. Disoriented crossings, knots, and links are defined with these knots embedding into a disoriented circle with 2n arcs, n N. The article gives examples and properties of the polynomials MK and NK and extends Kaufman polynomials F and L to disoriented link diagrams. This expansion of basic concepts, such as Reidemeister moves, linking numbers, and Kaufman’s bracket model, in addition to the polynomials of Jones, HOMFLY, and Kaufman, will aid future research in the study of disoriented knot theory