HOMFLY Polynomials of Pretzel Knots

TL;DR

This paper investigates HOMFLY polynomials of pretzel knots, identifying patterns and proposing conjectures that could lead to a general formula for these knot invariants, especially for pretzel knots with three odd parameters.

Contribution

It introduces initial steps toward a matrix-less general formula for HOMFLY polynomials of pretzel knots with three odd parameters, based on observed patterns and conjectures.

Findings

Patterns in HOMFLY polynomials for specific knots $9_{35}$ and $9_{46}$

Properties of $F$-factors in these polynomials

Conjectures on the structure of HOMFLY polynomials for pretzel knots

Abstract

HOMFLY polynomials are one of the major knot invariants being actively studied. They are difficult to compute in the general case but can be far more easily expressed in certain specific cases. In this paper, we examine two particular knots, as well as one more general infinite class of knots. From our calculations, we see some apparent patterns in the polynomials for the knots $9_{35}$ and $9_{46}$, and in particular their $F$-factors. These properties are of a form that seems conducive to finding a general formula for them, which would yield a general formula for the HOMFLY polynomials of the two knots. Motivated by these observations, we demonstrate and conjecture some properties both of the $F$-factors and HOMFLY polynomials of these knots and of the more general class that contains them, namely pretzel knots with 3 odd parameters. We make the first steps toward a matrix-less general formula for the HOMFLY polynomials of these knots.