A Symmetric Motion Picture of the Twist-Spun Trefoil

A 2-knot will be a 2-sphere embedded into R locally flatly. Although we cannot see a 2-knot directly with our eyes, there are several ways to visualize it. A motion picture is one of the ways. Roughly speaking, a motion picture is a CT scan (Computed Tomographic scan) of a 2-knot. It is a one-parameter family of slices of a 2-knot, that are 0or 1-dimensional objects in R. The n-twist-spun trefoil is a periodic 2-knot whose period is 2π/n (n ≥ 1). A motion picture of the twist-spun trefoil should exhibit the periodicity. However, we have not known such a symmetric motion picture so far. To obtain a symmetric motion picture, we have to see a whole shape of the twist-spun trefoil. It is difficult with paper and pencil. We thus consider to use a computer. In this paper, we first construct a symmetric diagram of the 2-twist-spun trefoil using a 3D modeling software. Here, a diagram is a projection image of a 2-knot by a projection R → R with over/under information. Slicing this diagram, we obtain a motion picture of the 2-twist-spun trefoil which exhibits the periodicity well. Observing this motion picture, we finally show the following theorem.