# Cyclotomic expansions of HOMFLY-PT colored by rectangular Young diagrams

## TL;DR

This paper proposes conjectures for closed-form expressions and cyclotomic expansions of HOMFLY-PT knot invariants colored by rectangular Young diagrams, using interpolation Macdonald polynomials, advancing the understanding of knot invariants.

## Contribution

It introduces conjectures for explicit formulas and cyclotomic expansions of HOMFLY-PT invariants for double twist and arbitrary knots with rectangular Young diagram colorings.

## Key findings

- Conjectured closed-form expressions for double twist knots.
- Proposed cyclotomic expansion formulas for general knots.
- Utilization of interpolation Macdonald polynomials in knot invariants.

## Abstract

We conjecture a closed-form expression of HOMFLY-PT invariants of double twist knots colored by rectangular Young diagrams where the twist is encoded in interpolation Macdonald polynomials. We also put forth a conjecture of cyclotomic expansions of HOMFLY-PT polynomials colored by rectangular Young diagrams for any knot.