For an easily-coded fabric working order, work along the boundary of a disk

A Jordan curve is a simple (i.e., non-self-intersecting,) closed curve in the plane. The Jordan Curve Theorem proves that such a curve divides the plane into two distinct regions. The fact that a Jordan curve keeps its two sides thoroughly walled off from one another is a fact tremendously useful to mathematics and computer science.

On closed surfaces, an analogy to a Jordan curve is a disk boundary, i.e., a curve that is the boundary of a topological disk. A disk boundary divides the surface into two distinct regions: the disk, which we consider the interior of the curve, and the exterior which is the remainder of the surface. The fact that a disk boundary walls off its interior from its exterior simplifies the coding of fabric working order on closed surfaces. Our working orders will always be disk boundaries.

Traditional fabric working orders generally do not follow disk boundaries.

Two traditional fabric working orders that do not follow a disk boundary—they are not even closed curves.

A traditional fabric working order familiar in jewelry making. Though it is a closed curve, its interior is not a disk.

Here are examples of fabric working orders that do follow disk boundaries: they end where they began, and they have an interior that is a topological disk.

Example of a fabric working order that follows a disk boundary.

Example of a fabric working order that follows a disk boundary.

Example of a fabric working order (the Moore curve) that follows the boundary of a disk,

The disk is easier to see if we color it:

A disk-boundary fabric working order with the disk colored gray.

A disk-boundary fabric working order with the disk colored gray.

A disk-boundary fabric working order (Moore's curve) with the disk colored gray.

The exterior of the disk boundary is always a closed surface less a disk. The exterior of the disk boundary may be quite complicated, but topologically it is always equivalent to a polygonal schema: a polygon with an even number of sides, which sides are to be paired up (identified), 2-by-2, in the correct orientation. Clearly, we can fill the interior of a polygon of any number of sides with a fabric working order like those illustrated above; for simplicity, we draw only a 4-sided polygonal schema representing a torus and show only the Moore curve's working order.

A fabric working order with a polygonal schema for a 1-hole torus as its exterior.

The edges of the polygonal schema are to identified in the way that matches labels and arrow directions. Working around the disk boundary in a counterclockwise direction, polygon edges are always on the right, and therefore topologically significant fabric connections are always right-side connections. No left-side connections cross the polygon edges; some right-side connections cross the polygon edges. In the Figure below: four right-side fabric connections that do not cross schema edges (shown in green,) and two right-side fabric connections that do cross schema edges (shown in red.)

A 1-hole torus worked in a counterclockwise direction. In green, four right-side fabric connections that do not cross edges of the polygonal schema; in red, two that do.

 If we identify edges of the polygonal schema one pair at a time, after the first pair we have a cylinder, or in other words, a sphere with two boundaries that later will be identified. Clearly there was no topological magic in the first edge pairing, as we already know how to make a sphere. The topological magic must be in how the two boundaries pair up. The situation is the same for an n-hole torus: we can identify certain pairs of polygonal edges until we get a sphere with 2n boundaries that still need pairing up.

n-hole tower

sphere with 2n boundaries 

For example, cuts that reduce an n-hole torus to a sphere with 2n boundaries can be found by rearranging the surface into an n-hole 'tower', and slicing from the top nearly to the bottom (Figure above.) In this case, lines showing the two-by-two pairing of these boundaries, and the identification of pairs of points on the boundaries can be drawn on the surface with non-crossing lines (Figure below.)

Identified points on the 2n boundaries connected by non-crossing lines drawn on the surface.

In theory, this case is general, because, having drawn non-crossing lines joining the paired boundaries in one arrangement, we can homotopically rearrange the boundaries as we wish without causing lines drawn on the surface to cross. Perhaps this will not be practical in practice (see Figure below,) we may need a tame arrangement of handles in the surface we are trying to make.

Non-crossing lines after a homotopic rearrangement of the surface.