On the Jones Polynomial of Knot Types
DOI:
https://doi.org/10.5281/zenodo.21899369Keywords:
Jonel polynomial, knot theory, trefoil knot, figure-eight knot, cinquefoil knot, Chern-Simons theory, quantum knot invariants, Khovanov homologyAbstract
In the recent years, knot theory has become a tool in applied mathematics, covering knots in higher dimensions by considering n-dimensional spheres and m-dimensional Euclidean space. While mathematical knots in general can be broad-ranging, narrowing down concepts can create interesting subject matters that may help understanding knots in a more comprehensive way, for instance, knot invariants. These invariants are quantities defined for each knot. One of the most known and significant knot invariants is the Jones polynomial. For certain types of knot diagrams, such as trefoil knot, figure-eight knot, and cinquefoil knot, there are various efficient classical algorithms for evaluating their Jones polynomial. This paper explores the concept of Jones polynomial in the mathematical field of knot theory and investigates how it is assigned to a particular knot. In addition, it seeks to identify properties of Jones polynomial that are significant to the knot types utilized in this study. It also discusses its relationship to other topological theories and invariants such as Chern-Simons theory, quantum knot invariants, and Khovanov homology. The finding establishes that for the first knot types with up to five crossings, the Jones polynomial can be generated by applying the skein relation. Overall, the study contributes to the systematic presentation and deeper understanding of Jones polynomial as a significant tool in studying mathematical knots.
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