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**# Knot Theory in Five Dimensions

#### by

## Samuel J. Lomonaco, Jr.

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Figure 1.
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A movie of a trivially embedded 2-sphere in S^{4}

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Figure 2.
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A movie of a knotted 2-sphere in S^{4}

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Figure 3.
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A normalized movie of the example found in Figure 1.
All hyperbolic points have been pushed into the frame X_{0}
(called the key frame).

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Figure 4.
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A normalized movie of the example found in Figure 2.
All hyperbolic ponts have been pushedi into frame X_{0} (called key frame).

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Figure 5.
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Labeling scheme for hyperbolic points.

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

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

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Figure 8.
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Movie of movies of 3-knot ( S^{5}, kS^{3} ).

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Figure 9.
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Index 0 Morse singularity.

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Figure 10.
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Index 1 Morse Singularity.

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Figure 11.
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Index 2 Morse singularity.

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Figure 12.
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Index 3 Morse singularity.

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Figure 13.
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Labeling scheme for index 1 singularity.

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Figure 14.
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Labeling scheme for index 2 singularity.

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Figure 15.
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Key frame representation of the 3-knot (S^{5}, kS^{3}) given in Figure 8.

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

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

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Figure 18.
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Wirtinger generators for a presentation of the fundamental group p_{1}X
of the complement of 3-knot given in Figure 17

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

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Figure 20.
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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.