Alternating quadrisecant approximation conjecture for knots with nontrivial Conway polynomial

Let $\gamma:\mathbb{R}\to\mathbb{R}^3$ be an analytic $\mathbb{Z}$-periodic parameterization with nonvanishing derivative of a knot whose Conway polynomial has a nontrivial quadratic term. For every $\varepsilon>0$, choose sequences

$$t_1<t_2<\cdots<t_6<t_1+1,\qquad s_1<s_2<s_3<s_4<s_1+1.$$

The points $\gamma(t_1),\ldots,\gamma(t_6)$ should form a trefoil knot when connected cyclically by line segments; the points $\gamma(s_1),\gamma(s_3),\gamma(s_2),\gamma(s_4)$ should lie in this order on some line; and

$$\min_{j=1,2,3,4}\lvert\gamma(t_i)-\gamma(s_j)\rvert<\varepsilon\qquad (i=1,\ldots,6).$$

Alternating quadrisecant approximation conjecture. Such sequences exist for every $\varepsilon>0$. The conjecture would connect the existence of alternating quadrisecants with inscribed trefoil knots. The paper notes that this might permit the nontriviality assumption on the quadratic term of the Conway polynomial to be replaced by the nontriviality of the knot, but that stronger statement is not asserted here.