The Arf-invariant conjecture for twist knots and second-order Lissajous representations

A second-order Lissajous knot is a knot $K$ represented parametrically by

$$x(t)=\cos(n_1t+\phi_1),\qquad y(t)=\cos(n_2t+\phi_2),$$ $$z(t)=\cos(n_3t+\phi_3)+\cos(n_4t+\phi_4),$$

where $n_1,n_2,n_3,n_4$ are integer frequencies, $\phi_1,\phi_2,\phi_3,\phi_4$ are real phase shifts, and $0\leq t\leq 2\pi$. The Arf-invariant conjecture. Every twist knot with Arf invariant $1$ is second-order Lissajous. The authors note that twist knots are Lissajous exactly when their Arf invariant is zero, and conjecture that the remaining twist knots have second-order Lissajous representations.