This paper tries to give a partial topological interpretation to the Jones polynomial. As the author remarks, this topological/geometrical meaning is better understood when the Jones polynomial is viewed as part of the coloured Jones function \(\{ J_K(n,q) \} _n\), and one looks at some suitable limits of this sequence.
For each natural number \(n\), the coloured Jones function of a knot \(K\) is given by a Laurent polynomial \(J_K(n,q)\) in \(q\). For \(n=2\) we recover the classical Jones polynomial. Let \(\delta _K^*(n)\) and \(\delta_K(n)\) be respectively the lowest and the highest degrees of this polynomial. By definition, the Jones slopes of \(K\) are the cluster points (limit points of a subsequence) of the sequences \(\{ \frac{4}{n^2}\delta_K^*(n)\}\) and \(\{ \frac{4}{n^2}\delta_K(n)\}\).
Let \(M_K\) be the exterior of a knot \(K\). Each properly embedded, essential (incompressible and boundary incompressible) orientable surface \(S\) in \(M_K\) determines some boundary curves in the torus \(T=\partial M_K\). Each of these boundary curves goes round \(a\) meridians and \(b\) longitudes of \(T\). By definition \(a/b \in \mathbb Q \cup \infty\) is said to be a boundary slope of the knot \(K\). In [Pac. J. Math. 99, 373–377 (1982; Zbl 0502.57005)], A. E. Hatcher proved that every knot has finitely many boundary slopes.
The author conjectures that, for any knot, its Jones slopes are always boundary slopes. In the paper the conjecture is proved for alternating knots, knots with at most nine crossings, torus knots and \((-2,3,n)\) pretzel knots.
Specifically, in the case of an alternating knot \(K\) with \(c^+\) positive crossings and \(c^-\) negative crossings in a reduced planar projection of \(K\), Theorem 2 states that the Jones slopes are \(-2c^-\) and \(2c^+\), and these are the boundary slopes of \(K\) that correspond to the two checkerboard surfaces of \(K\) (doubled, if needed, to make them orientable).
A relevant fact when studying the Jones slopes is that the coloured Jones function is holonomic (i.e., it satisfies a linear recursive relation). This implies (Theorem 1) that the sequence of degrees \(\{ \delta_K(n) \}\) is a quadratic quasi-polynomial for large \(n\). The Jones period of a knot is defined as the period of this quasi-polynomial (for the definition of the period of a quasi-polynomial see Definition 1.5), and it is seen to be one for every alternating knot. For knots with a non-integral Jones slope, this period is greater that one.
We remark that, in [D. Futer, E. Kalfagianni and J. S. Purcell, Proc. Am. Math. Soc. 139, No. 5, 1889–1896 (2011; Zbl 1232.57007)], the conjecture has also been proved for adequate knots.