On the quantum $\mathfrak{sl}_3$ invariant of positive links

TL;DR

This paper investigates the quantum rak{sl}_3 invariant of positive links using skein theory, providing explicit formulas for polynomial coefficients, characterizing fibered links, and identifying obstructions to positive braid representations.

Contribution

It introduces explicit formulas for the leading terms of the quantum rak{sl}_3 polynomial for positive links and relates polynomial coefficients to topological properties.

Findings

A positive link is fibered iff the second polynomial coefficient equals one.

The third coefficient obstructs positive braid representations.

Explicit diagrammatic formulas for polynomial coefficients.

Abstract

We use the skein theory of $\mathfrak{sl}_3$-webs to study the properties of the quantum $\mathfrak{sl}_3$-link polynomial of positive links. We give explicit formulae for the three leading terms of the polynomial on positive links in terms of diagrammatic quantities of their positive diagrams. We show that a positive link is fibered if and only the second coefficient of the polynomial is equal to one. We also show that the third coefficient provides obstructions to representing links by positive braids.