Chromatic Polynomial, Colered Jones Function and q-Binomial Counting

2004/12/22 by Martin Loebl, Loebl, Martin
Mathematics · #05A30 #57N10 #Combinatorics (math.CO) #FOS: Mathematics #Quantum Algebra (math.QA) #math.CO #math.QA #msc:05A30 #msc:57N10

paper · pdf · doi:10.48550/arxiv.math/0412460

arxiv created 2004/12/22 · arxiv updated 2009/12/01

Abstract

We define a q-chromatic function on graphs, list some of its properties and provide some formulas in the class of general chordal graphs. Then we relate the q-chromatic function to the colored Jones function of knots. This leads to a curious expression of the colored Jones function of a knot diagram as a 'defected chromatic operator' applied to a power series whose coefficients are linear combinations of chord diagrams constructed from 'flows' on the reduced knot diagram.

Related

• Random walks and the colored Jones function
• Foundations of the Colored Jones Polynomial of singular knots
• Difference and differential equations for the colored Jones function
• On the head and the tail of the colored Jones polynomial
• Generating functions of q-chromatic polynomials