On the quantum \mathfraksl₃ invariant of positive links

2025/08/21 by Matthew Harper, Harper, Matthew, Efstratia Kalfagianni +1
Mathematics · #57K10 #57K14 #57K16 (primary) #Advanced Operator Algebra Research #Algebraic structures and combinatorial models #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Quantum Algebra (math.QA)

paper · pdf · doi:10.48550/arxiv.2508.15153

openalex publication_date 2025/08/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We use the skein theory of \mathfraksl₃-webs to study the properties of the quantum \mathfraksl₃-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.

Citations

• On unknotting fibered positive knots and braids
• On the head and the tail of the colored Jones polynomial
• Closed incompressible surfaces in the complements of positive knots

Related

• A presentation for the category of sl(3) webs and an extension of the quantum sl(3)-invariant to tangles
• Skein theory for the Links-Gould polynomial
• Links, Quantum Groups, and TQFT's
• Tied Links
• A simple algorithm to compute link polynomials defined by using skein relations