Knots, Braids, and Knot Invariants

TL;DR

This paper introduces fundamental concepts in knot theory, reviews key invariants like the Jones and HOMFLY polynomials, and presents original research on computing the HOMFLY polynomial for specific braid closures, linking to future research directions.

Contribution

The paper provides a concise overview of knot invariants and introduces a new formula for the HOMFLY polynomial of looped coxeter braid closures, advancing computational methods in knot theory.

Findings

Derived a general formula for the HOMFLY polynomial of looped coxeter braids

Connected knot invariants to algebraic structures like Hecke algebras

Outlined future research on universal trace in knot invariants

Abstract

In this report, I will start by first giving a brief introduction on knots to build some intuition before beginning the more rigorous review in the Literature Review section. There, I will define knot equivalence, the Jones polynomial invariant, braid groups, Hecke algebras, and the HOMFLY polynomial invariant. Next, I will cover some of my independent research regarding the computation of a general formula for the HOMFLY polynomial of the closure of looped coxeter braids. Lastly, I will loosely discuss how this research connects to future research regarding the universal trace in the Plan for Future Work section.