More on knots in Robinson's attractor (Q2033196)

A {knot} is a smooth oriented embedding of \(S^1\) into \(\mathbb{R}^3\), and a {link} is a finite set of knots with disjoint images. In this paper, following [\textit{M. C. Sullivan}, J. Knot Theory Ramifications 7, No. 1, 115--121 (1998; Zbl 0948.57004)], the authors are interested in knotted periodic orbits of the following system of ordinary differential equations: \begin{align*} &\dot{x} = y,\\ & \dot{y} = x-2x^3+\alpha y + \beta x^2y+yz,\\ &\dot{z} = -\gamma z + \delta x^2, \end{align*} where \(\alpha = -0.71\), \(\beta = 1.8690262\), \(\delta = 0.1\) and \(\gamma=0.6\). They show that the knots and links are fibered (i.e., if the complement of a tubular neighborhood in \(S^3\), taken as the one-point compactification of \(\mathbb{R}^3\), can be fibered over \(S^1\) with fiber an orientable surface), with positive signature (i.e., it or its mirror image has a knot diagram with only positive crossings) and that each knot-type appears infinitely often. Moreover, they construct a zeta type function that counts periodic orbits by the twisting of the local stable manifolds, with a notion of twisting that is not the usual one.