Knot Theory in Five Dimensions
Samuel J. Lomonaco, Jr.
A movie of a trivially embedded 2-sphere in S4
A movie of a knotted 2-sphere in S4
A normalized movie of the example found in Figure 1.
All hyperbolic points have been pushed into the frame X0
(called the key frame).
A normalized movie of the example found in Figure 2.
All hyperbolic ponts have been pushedi into frame X0 (called key frame).
Labeling scheme for hyperbolic points.
Key frame representation of example found in Figures 1 and 3.
Key frame representation of example found in Figures 2 and 4.
Movie of movies of 3-knot ( S5, kS3 ).
Index 0 Morse singularity.
Index 1 Morse Singularity.
Index 2 Morse singularity.
Index 3 Morse singularity.
Labeling scheme for index 1 singularity.
Labeling scheme for index 2 singularity.
Key frame representation of the 3-knot (S5, kS3) given in Figure 8.
A non-standard movie of movies of 3-knot in Figures 8 and 15.
A non-standard single frame representation of the 3-knot given in Figures 8, 15, and 16.
Wirtinger generators for a presentation of the fundamental group p1X
of the complement of 3-knot given in Figure 17
Representation of the general aspherical decomposition of the complement of a 3-knot.
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.