Fig. 3

A space curve homotopy where the helical segment moves upwards and the straight segment moves inclined forward. The first intersection of the two segments is shown in blue and the second in green. Following the motion of the point of intersection on the blue helical segment between the two times of intersection, you get the black cylinder to the right-hand side. Inside this black cylinder the green helical segment can go down and around the intersection point of the two blue curve segments and back up to the green curve. Similarly, the gray cylinder to the right-hand side shows the motion of the first intersection point on the straight segment. Inside the gray cylinder, the green line segment may be altered to go back and around the intersection point. Hence, if the right hand side black and gray cylinders do not intersect other parts of the curve at a time between the two times of intersection, then we can alter all curves inside the two cylinders and postpone the intersection. Similarly, if the black and gray cylinders to the left-hand side do not intersect other parts of the curve between the times of the two intersections, then that intersection can be mover forward in time. Now move both intersections to happen at the mean of the two original intersection times (shown in red). The two red arcs combined with parts of the black and gray cylinders now form a closed loop. If a topological disc spanned by this loop avoids the remaining parts of the red curve, then two Reidemeister moves of type 2 can change the initial under-sliding of the helical segment to an over-sliding and avoid the two intersection points