Higher order terms in the Melvin-Morton expansion of the colored Jones polynomial. (English) Zbl 0882.57004
In [ibid. 169, No. 3, 501-520 (1995; Zbl 0845.57007)] P. M. Melvin and H. R. Morton considered the expansion of the coloured Jones function for a knot as a power series in the variable \(h\) (with \(q=e^h)\) where the coefficients depend on the ‘colour’ \(k\), in other words the dimension of the irreducible representation of the quantum group \(SU(2)_q\) used to colour the knot. They showed that in this expansion each coefficient is a polynomial in \(k\). They conjectured that the coefficient of \(h^n\) had degree not greater than \(n\) and that taking only the term of degree \(n\) in \(k\) from the coefficient of \(h^n\) would produce a power series derived from the inverse of the Alexander polynomial of the knot. This conjecture was proved by D. Bar-Natan and S. Garoufalidis [Invent. Math. 125, No. 1, 103-133 (1996; Zbl 0855.57004)], while the author himself gave an earlier path-integral account in [Commun. Math. Phys. 175, No. 2, 275-296 (1996; Zbl 0872.57010)].
In the present paper the author writes the coloured Jones expansion in terms of variables \(h\) and \(z= e^{kh}-e^{-kh}\) to get \(\sum V^{(n)} (z)h^n\). In this form the highest degree terms of the Melvin-Morton expansion are given simply by \(n=0\), and the Melvin-Morton conjecture can be written as \(V^{(0)} (z)=1/ \nabla (z)\), where \(\nabla (z)\) is the Conway polynomial of the knot. The author considers here the nature of the coefficients \(V^{(n)} (z)\) as a function of \(z\) for larger values of \(n\). He conjectures that the product \(V^{(n)} (z) (\nabla(z))^{2n+1}\) is always an even polynomial in \(z\) with integer coefficients. These functions, for varying \(n\), give the ‘lines’ in the Melvin-Morton power series in which the degree of the coefficient polynomials falls short of its maximum by \(n\).
The conjecture is proved in this paper for torus knots, and experimental and heuristic evidence for the general result is presented. A note added in proof refers to a later preprint by the author in which the conjecture is established in general using properties of the universal \(R\)-matrix for \(SU(2)_q\).
The careful reader should be aware that the parameter \(h\) used by Melvin and Morton is not quite the same as that used here. In effect the author uses \(h=q-1\) in place of \(q=e^h\). The general nature of the power series expansions are not, however, significantly affected by this.
In the present paper the author writes the coloured Jones expansion in terms of variables \(h\) and \(z= e^{kh}-e^{-kh}\) to get \(\sum V^{(n)} (z)h^n\). In this form the highest degree terms of the Melvin-Morton expansion are given simply by \(n=0\), and the Melvin-Morton conjecture can be written as \(V^{(0)} (z)=1/ \nabla (z)\), where \(\nabla (z)\) is the Conway polynomial of the knot. The author considers here the nature of the coefficients \(V^{(n)} (z)\) as a function of \(z\) for larger values of \(n\). He conjectures that the product \(V^{(n)} (z) (\nabla(z))^{2n+1}\) is always an even polynomial in \(z\) with integer coefficients. These functions, for varying \(n\), give the ‘lines’ in the Melvin-Morton power series in which the degree of the coefficient polynomials falls short of its maximum by \(n\).
The conjecture is proved in this paper for torus knots, and experimental and heuristic evidence for the general result is presented. A note added in proof refers to a later preprint by the author in which the conjecture is established in general using properties of the universal \(R\)-matrix for \(SU(2)_q\).
The careful reader should be aware that the parameter \(h\) used by Melvin and Morton is not quite the same as that used here. In effect the author uses \(h=q-1\) in place of \(q=e^h\). The general nature of the power series expansions are not, however, significantly affected by this.
Reviewer: H.R.Morton (Liverpool)
MSC:
| 57M25 | Knots and links in the \(3\)-sphere (MSC2010) |
References:
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