Chromatic Polynomial, Colored Jones Function and Q-binomial Counting
نویسندگان
چکیده
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 K as a ’defected chromatic operator’ applied to a power series whose coefficients are linear combinations of chord diagrams constructed from ’flows’ on reduced K.
منابع مشابه
Difference and Differential Equations for the Colored Jones Function
The colored Jones function of a knot is a sequence of Laurent polynomials. It was shown by TTQ. Le and the author that such sequences are q-holonomic, that is, they satisfy linear q-difference equations with coefficients Laurent polynomials in q and qn. We show from first principles that q-holonomic sequences give rise to modules over a q-Weyl ring. Frohman-Gelca-LoFaro have identified the latt...
متن کاملThe colored Jones function is q–holonomic
A function of several variables is called holonomic if, roughly speaking, it is determined from finitely many of its values via finitely many linear recursion relations with polynomial coefficients. Zeilberger was the first to notice that the abstract notion of holonomicity can be applied to verify, in a systematic and computerized way, combinatorial identities among special functions. Using a ...
متن کاملRandom Walks And The Colored Jones Function
It can be conjectured that the colored Jones function of a knot can be computed in terms of counting paths on the graph of a planar projection of a knot. On the combinatorial level, the colored Jones function can be replaced by its weight system. We give two curious formulas for the weight system of a colored Jones function: one in terms of the permanent of a matrix associated to a chord diagra...
متن کاملRandom Walk on Knot Diagrams, Colored Jones Polynomial and Ihara-selberg Zeta Function
A model of random walk on knot diagrams is used to study the Alexander polynomial and the colored Jones polynomial of knots. In this context, the inverse of the Alexander polynomial of a knot plays the role of an Ihara-Selberg zeta function of a directed weighted graph, counting with weights cycles of random walk on a 1-string link whose closure is the knot in question. The colored Jones polyno...
متن کاملAsymptotics of the Colored Jones Polynomial and the A-polynomial
The N-colored Jones polynomial JK (N) is a quantum invariant which is defined based on the N-dimensional irreducible representation of the quantum group Uq(sl(2)). Motivated by Volume Conjecture raised by Kashaev [16], it was pointed out that the colored Jones polynomial at a specific value should be related to the hyperbolic volume of knot complement [21]. As another example of the knot invari...
متن کامل