On knot invariants - Earnhart, Richard T. (Animo Repository, 1999)
This study is based primarily on the papers New Invariants in the Theory of Knots by L.H. Kauffman, American Mathematical Monthly, Vol. 95, No. 3, March 1988 and The Color Invariants of Knots and Links by P. Andersson, American Mathematical Monthly, Vol. 102, No. 5, May 1995, presents the most elementary concepts of knots theory such as knot types and definitions, diagram moves, linking, writing and twisting numbers. It used combinatorial methods to present some variants of knots and links with emphasis on Kauffman's bracket polynomial and the new combinatorial approach introduced by Andersson for the colorability of knots and links. The bracket invariant is constructed and is used to prove knottedness and nontriviality of some knots and links and the chirality of the trefoil knot. The bracket invariant is also used to determine properties of alternating knots and links. One of these properties is the invariance in the degree of polynomial of a knot. This study also discusses the equivalence of the colorability mod n of a knot or link diagram to solving a system of linear equations. Several examples and illustrations are provided to determine the colorability or knottedness of some knots and links. The topological invariants of knots and links presented in this study have used only diagrams and few calculations that make this study of knots simple but very powerful and interesting. Most of the invariants discussed specially those of bracket and color invariants follow directly from the three fundamental moves or better known as Reidemeister moves. The bracket polynomial K is a regular isotopy, that is, it cannot be affected by Type II and Type III moves. If Type I move is to be applied, the bracket K is multiplied by alpha -w(k). The polynomial obtained is called the f-polynomial and this is unaffected by any of the three fundamental moves. Properties of the f-polynomial can be used to prove knottedness and chirality of different knots or links. Another combinatorial tool of proving knottedness and chirality of a knot or link is the colorability mod n. This concept of colorability is equivalent to solving a system of linear equations. A knot can be colored mod n if there is an integer solution to the system of linear equations representing a knot. The study of knots is a rapidly developing field of research with many applications in DNA, topological stereochemistry and in statistical mechanics.