A Weight System Derived from the Multivariable Conway Potential Function

In [3] D. Bar-Natan and S. Garoufalidis used a weight system for the Alexander polynomial of knots to prove the so-called Melvin–Morton–Rozansky conjecture [18, 21] which relates the Alexander polynomial and some coefficients of coloured Jones polynomial. Their weight system can be easily extended to the case of links. It is a natural problem to give a weight system for the multivariable Alexander polynomial or its normalised version, the Conway potential function for links. The Conway potential function was first introduced by J.H. Conway [6] by giving some ‘axioms’. Unfortunately, his ‘axioms’ are not sufficient to define his potential function. R. Hartley [11] gave its precise definition by using R.H. Fox’s free differential calculus [9, 10]. He also showed that for two-bridge links Conway’s first two identities and initial data for the trivial knot, for split links, and for the positive Hopf link are sufficient to calculate the potential function. After that M.E. Kidwell [13] proved they are also sufficient for calculation of links with two labels K = T ∪L where T is an unknotted circle and T and L have different labels. (Note that L may have more than one component.) Then Y. Nakanishi [20] proved that we can calculate the potential function if the number of labels (which equals the number of variables) is two or three. Besides Hartley’s axioms we need Conway’s third identity and initial data for the connected sum of two positive Hopf links and for the three-component positive Hopf link. Finally J. Murakami proved that Conway’s first and second identities, a connect sum formula for the positive Hopf link, initial data for the trivial knot and for split links, and his new relation involving seven locally distinct links are sufficient to calculate the Conway potential function for links with any number of labels. In this paper we use J. Murakami’s result to define a weight system. Moreover we will show that our weight system can be calculated recursively by using five axioms. Since the proof is fairly easy we expect that there may be similar weight systems. If so, by using M. Kontsevich’s integral [14] we could then define invariants for labelled links other than the Conway potential function. It is also an interesting problem whether our weight system is canonical, that is, whether we obtain the