Let $K \subset S^3$ be a knot which is both strongly invertible and periodic, that is, $K$ is fixed by both a smooth involution $\tau: S^3 \to S^3$ which preserves the orientation of $S^3$ but reverses the orientation of $K$, and $K$ is fixed by a finite order diffeomorphism $\rho: S^3 \to S^3$ which preserves the orientation of both $S^3$ and $K$ and which has a non-empty fixed-point set.

Many such knots exist, consider for example the figure-eight knot:

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In this diagram an order 2 period can be seen by $\pi$ rotation around a line perpendicular to the diagram, and a strong inversion can be seen by $\pi$ rotation around a vertical line. $9_{40}$ is a 3-periodic example:

_{[source]}

By the resolution of the Smith conjecture, a periodic or strongly invertible symmetry will have an unknotted fixed set. In these diagrams for $4_1$ and $9_{40}$, the fixed set of the period is an axis perpendicular to the diagram and the fixed set of the strong inversion is an axis contained in the plane of the diagram so that these axes intersect at the origin and infinity. Does every knot which is both strongly invertible and periodic have such a diagram?

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