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.