Exploring Invariants in Knot Theory

This thesis will explore the importance of invariants within the mathematical branch of Knot Theory with an emphasis on polynomial invariants and one interesting invariant within the subfield of tangle theory. The paper will begin by providing a basic foundation of mathematical knots and links that will be essential to understanding the discussion of invariants. The paper will then explain the importance of invariants and how they have and can be used to answer some of the most crucial questions in Knot Theory. This paper will largely focus on the Alexander Polynomial, the Conway Polynomial, the Jones Polynomial, and the HOMFLYPT Polynomial. For each polynomial, the paper will explain how it is calculated for both knots and links, discuss some important qualities, and analyze crucial limitations. At the end of this section, the paper will provide a discussion comparing and contrasting the relative values of each invariant over different types of knots. Furthermore, the paper will transition into Tangle Theory. I will begin by providing the fundamentals of tangles, emphasizing rational tangles, which is necessary for understanding the rest of the paper. Next, the paper will explain exactly how tangles can be transitioned into knots and links to tie the new concepts into the start of the paper. Finally, the paper will explore an interesting link invariant of tangle theory, specifically highlighting how it differs from the polynomial invariants in its bidirectionality. Lastly, the paper will give a short discussion on potential further explorations of invariants, highlighting goals, open questions, and struggles currently faced.