Alexander-Conway and Bracket Polynomials of Pretzel Links $\boldsymbol{P(1,1,n)}$

TL;DR

This paper investigates the Alexander-Conway and Kauffman bracket polynomials for a specific family of three-strand pretzel links, contributing to the understanding of polynomial invariants in knot theory.

Contribution

It provides explicit calculations and analysis of these polynomials for the pretzel links P(1,1,n), enhancing the understanding of their algebraic properties.

Findings

Explicit formulas for Alexander-Conway polynomials of P(1,1,n)

Explicit formulas for Kauffman bracket polynomials of P(1,1,n)

Insights into the polynomial behavior for varying n

Abstract

Polynomial invariants constitute a dynamic and essential area of study in the mathematical theory of knots. From the pioneer Alexander polynomial, the revolutionary Jones polynomial, to the collectively discovered HOMFLYPT polynomial, just to mention a few, these algebraic expressions have been central to the understanding of knots and links. The introduction of Khovanov homology has sparked significant interest in the categorification of these polynomials, offering deeper insights into their topological and algebraic properties. In this work, we revisit two prominent polynomial invariants, the Alexander-Conway and the Kauffman bracket polynomials, and focus specifically on the polynomials associated with the family of three strand pretzel links $P(1,1,n)$.