Knot Theory in Five Dimensions

by

Samuel J. Lomonaco, Jr.

  • Figure 1.
    A movie of a trivially embedded 2-sphere in S4

  • Figure 2.
    A movie of a knotted 2-sphere in S4

  • Figure 3.
    A normalized movie of the example found in Figure 1. All hyperbolic points have been pushed into the frame X0 (called the key frame).

  • Figure 4.
    A normalized movie of the example found in Figure 2. All hyperbolic ponts have been pushedi into frame X0 (called key frame).

  • Figure 5.
    Labeling scheme for hyperbolic points.

  • Figure 6.
    Key frame representation of example found in Figures 1 and 3.

  • Figure 7.
    Key frame representation of example found in Figures 2 and 4.

  • Figure 8.
    Movie of movies of 3-knot ( S5, kS3 ).

  • Figure 9.
    Index 0 Morse singularity.

  • Figure 10.
    Index 1 Morse Singularity.

  • Figure 11.
    Index 2 Morse singularity.

  • Figure 12.
    Index 3 Morse singularity.

  • Figure 13.
    Labeling scheme for index 1 singularity.

  • Figure 14.
    Labeling scheme for index 2 singularity.

  • Figure 15.
    Key frame representation of the 3-knot (S5, kS3) given in Figure 8.

  • Figure 16.
    A non-standard movie of movies of 3-knot in Figures 8 and 15.

  • Figure 17.
    A non-standard single frame representation of the 3-knot given in Figures 8, 15, and 16.

  • Figure 18.
    Wirtinger generators for a presentation of the fundamental group p1X of the complement of 3-knot given in Figure 17

  • Figure 19.
    Representation of the general aspherical decomposition of the complement of a 3-knot.

  • Figure 20.
    Another representation of the general aspherical decomposition of the complement of a 3-knot. Arrors denote inclusions. For clarity, the piece INFINITY together with its arrows is not shown.