Invariants of knots in geodesic flows

The invariant $M_3$ of 3-component links is introduced in [2]. A new invariant $M_5$ of oriented 5-component links with cyclic order of components is introduced. Invariants have asymptotic and ergodic properties. An expression of the invariant $M_3$ by means of coefficients of the Conway polynomial is in [3].
The invariant $M_3$ is calculated for knotted trajectories of ergodic flows on the lens space $\mathbb S^3/\mathbb Z_3$. The flows are defined using the geodesic flow on the Lobachevskii plane with the modular group symmetry. The spaces of the flows have a common volume, metrics depend on a curvature. Analogical calculation for Arnold’s asymptotic ergodic linking numbers [1] is in [4].
References
[1] Arnold, V. I. 1974 The asymptotic Hopf invariant and its applications, Sel. Math. Sov. 5, 327–345.
[2] Akhmet’ev, P.M. On a higher integral invariant for closed magnetic lines, Journal of Geometry and Physics 74 (2013) 381–391.
[3] Akhmet’ev, P.M. On combinatorial properties of a higher asymptotic ergodic invariant of magnetic lines, Journal of Physics: Conference Series 544 (2014) 012015.
[4] P.M.Akhmet’ev, S.Candelaresi and A.Y.Smirnov. Minimum quadratic helicity states, J. Plasma Phys., (2018) 1–16.