On universal knot polynomials

TL;DR

This paper introduces universal knot polynomials for 2- and 3-strand torus knots in the adjoint representation, unifying various group-specific polynomials and exploring their invariance and potential for universalization across all knots.

Contribution

It develops a universal formulation of knot polynomials in the adjoint representation, connecting different Lie groups and proposing a universal approach for all knots.

Findings

Universal polynomials match known HOMFLY and Kauffman polynomials on specific lines.

Topological invariance [m,n]=[n,m] holds across Vogel's plane.

Proposes a universal form for the figure-eight knot invariant.

Abstract

We present a universal knot polynomials for 2- and 3-strand torus knots in adjoint representation, by universalization of appropriate Rosso-Jones formula. According to universality, these polynomials coincide with adjoined colored HOMFLY and Kauffman polynomials at SL and SO/Sp lines on Vogel's plane, and give their exceptional group's counterparts on exceptional line. We demonstrate that [m,n]=[n,m] topological invariance, when applicable, take place on the entire Vogel's plane. We also suggest the universal form of invariant of figure eight knot in adjoint representation, and suggest existence of such universalization for any knot in adjoint and its descendant representation. Properties of universal polynomials and applications of these results are discussed.