Basic concepts of disoriented knot theory

The aim of this thesis is to provide a basis for the disoriented knot theory by revealing the diagrammatic and polynomial invariants of disoriented knots and links. For this purpose, the concept of disoriented knot and link has been redefined. Thanks to this new definition, which generalizes the definition of a disoriented knot given for the first time in [1], the connected sum of disoriented knots is easily explained, and the number of all disoriented diagrams of a knot has been determined. For the disoriented knot and link diagrams, the minimum set of generators that produce all Reidemeister moves has been determined. This minimum set of generators has been used as a baseline method to prove invariants of disoriented knots and links. In this context, firstly, Gauss codes have been defined for disoriented knot and link diagrams, and Gauss diagrams have been drawn on disoriented circles with the help of Gauss codes. The Gauss code corresponding to each move of the minimum generator set of the disoriented Reidemester moves has been determined, and Gauss diagrams have been drawn. After revealing the diagrammatic invariants of disoriented knots and links, the subject of polynomial invariants, which is very important in the classification of disoriented knots and links, has been discussed. In this regard, a bivariate polynomial M, which is a generalization of the bracket polynomial for disoriented knot and link diagrams, has been defined and proved to be a well-defined regular isotopic invariant. This polynomial has extended the Kauffman L polynomial, which is the important invariant of the regular isotopy for classical knot and link diagrams, to disoriented knot theory. The polynomial M has been normalized by complete writhe to obtain a polynomial called N, which is an invariant of the ambient isotopy for disoriented link diagrams. It shows that this normalized polynomial is both a generalization of the Jones polynomial for disoriented links and extends the Kauffman F polynomial in classical knot theory to disoriented link diagrams. At the same time, some basic properties of M and N polynomials have been proved, and the polynomials M and N of several disoriented knot and link diagrams have been calculated. The foundations of the disoriented knot theory have been laid to a large extent, with two articles produced from this thesis and [1] published in journals and indexed in the ISI database. In this respect, this thesis and the articles produced will be a source for the development of knot theory.