Seams in synthetic weaving

Progressively building a higher genus surface (such as a torus) necessarily involves a seam, that is, a frontier where new work joins to older work. When the surface building technique is synthetic weaving, the seams will be invisible in the finished work because synthetic weaving is, in a sense, "all seam." The importance to synthetic weaving of seams has only to do with the order of weaving.

Opening all the seams should leave the work connected and planar: if we find the work is disconnected, unnecessary seams were made; if we find the work is still nonplanar, some unopened seams must remain in the work.

Opened up in this way, the work is a polygon with an even number of edges, two for each seam. This is referred to as a polygonal model or planar model in topology texts.

A planar model corresponds to a single-face map drawn on the surface. Each seam is represented by an edge in the map and two paired edges in the polygonal model. Each edge in the map necessarily borders the single face on both sides, and thus is encountered twice in a walk around the perimeter of the face. In an orientable surface the second passage along the same edge must be in the opposite direction of the first. In algebraic topology the sequence of edges encountered is described by an edge word, such as aba^-1b^-1, where the inverse indicates traversing the edge in the opposite direction.

For the orientable surfaces that concern us, as the paired edges will always join in the natural, untwisted orientation. Therefore, we can indicate the pairings by simply adding to the polygon diagram some curvy lines connecting edge pairs, as in the figure above.

Many ways of seaming are possible, but only some can encoded in undip efficiently by the addition of a specialized u and d. These are namely the seam designs that can be diagrammed on the planar model without crossing lines (for example, Canonical and Longitude/Latitude in the figure above.) The specialized u and d are only needed for bonds that violate planarity by trespassing on the regular weaving in the interior of the polygonal model.

Canonical starts and finishes one handle at a time. Opposite Sides has all the handles in progress at one time but results in a maximally non-planar diagram. Longitude/Latitude uses nearly twice as many edges as the other designs, has all handles in progress at one time, but is planar, so it can be encoded in undip with just the specialized u and d.

The Longitude/Latitude seam design uses more edges than Canonical, but may be more practical in many cases as it exploits multiple vertices. The dashed lines connect corresponding ends of corresponding edges. The dashed orbits reveal that the number of distinct vertices in the map is three—one fourth the number of edges in the polygon.

Undip codes on higher genus surfaces

Shuffled Dyck words code hamiltonian 3-regular maps on the sphere. These maps also correspond to synthetic weaving on the sphere, as the loop of wire corresponds to the hamiltonian circuit and each edge not in the circuit corresponds to the bond made by a pair of staples.

Surfaces of higher genus than the sphere can always be cut open into a planar polygon with 2n edges that are to be identified (i.e., paired up and glued back together when the time comes to reassemble the surface.) If the hamiltonian circuit never crosses the polygon, everything will be planar except, possibly, the few bonds that cross the polygon. These bonds may need some supplementary coding to be properly reconnected.

When the polygon in question is a square, the only two orientable surfaces it can represent are the sphere and the torus.

For the sphere, the polygon-crossing bonds reconnect in a way that, as one would expect, can be diagrammed on the plane without crossing lines.

Polygonal model of a spherical surface. Notice that when the closed hamiltonian circuit is traversed in the direction indicated, all polygon-crossing bonds are on the lefthand side (coded by u or d) and are visited in clockwise order.

 Therefore these polygon-crossing bonds are not in any way special and can be undip coded in the usual way along with all the other bonds that do not cross the polygon. Notice that, given the orientation of the hamiltonian circuit indicated all these bonds will be coded by u and d (there will be other bonds coded by u and d that do not cross the polygon, but no distinction need to be made.)

For the torus, the polygon-crossing bonds re-connect in a way that does not permit drawing their re-connection on the plane without crossing lines. However, as evidenced by the drawing below, the re-connection for either of the two cuts can be drawn on the plane, just not both at the same time.

Polygonal model of a torus. Either class of bonds, but not both, can be drawn in the plane without crossings. As drawn, only the pink bonds need special coding.

We could code the the torus above by introducing two new characters for bond connections we cannot draw on the plane. For example, we could use a pink u and a pink d. Pink u's and d's, connect up with each other in the same planar way, they are just, so to speak, drawn on a separate page.

Does a double torus, triple torus, etc., need more colors? The polygonal model for a double torus is an octagon, shown here by subdividing the edges of a square.

Polygonal model of a double torus. When the model is in canonical form, as here (i.e.,  aba'b'cdc'd') the edges fall into two classes ({a, c} and {b, d}), either of which can be diagrammed on the plane. 

A double torus can be cut into a polygon in various ways, but the canonical way gives an octagon with edges ordered a b a' b' c d c' d', as in the drawing above. These edges partition into two classes whose re-connections can be drawn in the plane, just not both classes at the same time. Here again, we just need a pink u and d to code this surface. These two additional characters also suffice for any higher genus orientable surface that has been dissected in the canonical way.

Actually, we do not always have to have a canonical dissection in order to get by with just two additional characters. It suffices to have a dissection that partitions the polygon-crossing edges into two sets, each of which can be described by a parenthesis word. For instance, in a canonical polygon, the parenthesis word for either set happens to be ()()()..., but for our purposes any parenthesis word would serve.

Tighter synthetic weaving joins and marking by bend type

The tightest synthetic weaving join occurs when a double-bump bend (follow upper black wire) accommodates a single-bump bend (follow lower black wire.) (The edges shown have not yet been locked by the overlaying of third strands.)

Making the tightest possible join at a synthetic weave's 'crossings' adds a slight complexity since complementary portions (as opposed to congruent portions) must be brought into alignment. On the other hand, these two complementary types of bends can be exploited to indicate 'open' or 'close', without any other sort of marking needed on the scaffold strand.

The basket will still contain edges of both even and odd lengths (as measured in helical half wavelengths.)

A scaffold strand for a synthetic weave can be described by a four letter code (u, n, d, p). This is an 'undip' word as described earlier in this blog, only now the 'photon' edges have shrunk to zero length. Two sequential undip letters can be 'alike' or 'unlike' in two comparisons: up/down and open/closed.

Edges bounded by letters that compare as alike/alike or unlike/unlike will be even in length.

Edges bounded by letters that compare as alike/unlike or unlike/alike will be odd in length.

More concisely, call letter pairs {u, p} and {d, n} opposites. Then an edge has even length if it is bounded by identical letters or by opposite letters. Otherwise, it has odd length.

Same join as above after overlaying third strands.