On the geometry of two state models for the colored Jones polynomial

Abstract

Using the flow property of the RR-matrix defining the colored Jones polynomial, we establish a natural bijection between the set of states on the part arc-graph of a link projection and the set of states on a corresponding bichromatic digraph, called arc-graph, as defined by Garoufalidis and Loebl [A non-commutative formula for the colored Jones function, Math. Ann. 336 (2006) 867–900]. We use this to give a new and essentially elementary proof for the knot state-sum formula in [S. Garoufalidis and M. Loebl, A non-commutative formula for the colored Jones function, Math. Ann. 336 (2006) 867–900]. We will show that the state-sum contributions of states on the part arc-graph defined by the universal RR-matrix of Uq(sl(2,C))Uq(sl(2,ℂ)) correspond, under our bijection of sets of states, to the contributions in [S. Garoufalidis and M. Loebl, A non-commutative formula for the colored Jones function, Math. Ann. 336 (2006) 867–900]. This will show that the two state models are in fact not essentially distinct. Our approach will also extend the formula of Garoufalidis and Loebl to links. This requires some additional nontrivial observations concerning the geometry of states on the part arc-graphs. We will discuss in detail the computation of the arc-graph state-sum, in particular for 33-braid closures