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# 数学代写|扭结理论代写Knot Theory代考|MATH148 Shrink the knot to ideal vertices for the bottom polyhedron

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## 数学代写|扭结理论代写Knot Theory代考|Shrink the knot to ideal vertices for the bottom polyhedron

Notice that underneath the knot, the picture of faces, edges, and vertices will be slightly different. In particular, when finding the top polyhedron, we collapsed overstrands to a single ideal vertex. When you put your head underneath the knot, what appear as overstrands from below will appear as understrands on the usual knot diagram.

One way to see this difference is to take the 3-dimensional model constructed in Figure 1.4. Figure 1.3 shows the view of the faces meeting at an edge from the top. If you turn the model over to the opposite side, you will see how the faces meet underneath. Figure $1.9$ illustrates this. Note that $U$ now meets $V$, and $S$ meets $T$.

In terms of the combinatorics, edges of Figure $1.5$ that are isotopic by sliding an endpoint along an understrand are identified to each other on the bottom polyhedron, but edges only isotopic by sliding an endpoint along an overstrand are not identified.

As above, collapse each knot strand corresponding to an understrand to a single ideal vertex. The result is Figure 1.10.

One thing to notice: we sketched the top polyhedron with our heads inside the ball on top, looking out. If we move the face $D$ away from the point at infinity, then it wraps above the other faces shown in Figure 1.8.
On the other hand, we sketched the bottom polyhedron with our heads outside the ball on the bottom. If we move the face $D$ away from the point at infinity, it wraps below the other faces shown in Figure 1.10.

## 数学代写|扭结理论代写Knot Theory代考|Rebuilding the knot complement from the polyhedra

1.1.5. Rebuilding the knot complement from the polyhedra. Figures $1.8$ and $1.10$ show two ideal polyhedra that we obtained by studying the figure- 8 knot complement. We claim that they glue to give the figure- 8 knot complement. That is, attach face $A$ on the bottom polyhedron to the face labeled $A$ on the top polyhedron, ensuring that the edges bordering face $A$ match up. Similarly for the other faces.

This process of gluing faces and edges gives exactly the complement of the knot. By construction, faces glue to give the faces illustrated in Figure 1.6, and edges glue to give the edges there, except that when we have finished, all four edges in an isotopy shown in that figure have been glued together.

## 数学代写|扭结理论代写Knot Theory代考|Rebuilding the knot complement from the polyhedra

1.1.5。从多面体重建结补。数字1.8和1.10显示我们通过研究图 8 结补获得的两个理想多面体。我们声称它们粘合以提供数字 8 结补码。也就是贴脸一个在底部多面体上标记的面一个在顶部多面体上，确保边缘与面接壤一个配对。其他面孔也是如此。

## MATLAB代写

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