Guan Keying

Department of Mathematics, School of Science, Beijing Jiaotong University

Email:  keying.guan@gmail.com

Abstract: It is pointed out that in addition to the kink structure, a smooth closed space curve has another important differential topological invariant, namely the number of rotations (circles) around a certain axis. Only by combining mathematics and physics can the concept of the number of rotations be accurately described.

1. Physical background and mathematical problems

The author published a blog post on ScienceNet in 2016 

• Mobius      strips, kinks and period-doubling bifurcations of spatial limit cycles -      Research progress  ( 12527 views ) Popularity  2  2015-4-24      17

Later, in 2018, the author further proposed the concept of the number of rotations (circles) of closed curves in space on ScienceNet.

Differential Topological Invariant of Frenet Closed Curve in Space--Rotation Number

Because my computer hard drive was damaged at the time, I could not find the detailed papers. I was also focusing on other issues and became lazy during the long pandemic, so my research in this area was put on hold. Until recently, I noticed that this research was directly related to my recent physics research on "static force also does work" and "equivalence of power and force", so I summarized my research results in more detail and wrote this blog post.

To help readers intuitively understand the physical and mathematical background of this problem, the author attaches several common photos in life:

Figure 1. Water flosser

Figure 1 is an electric water flosser that has been used for many years. Note that there is a PVC water pipe with six turns under the toothbrush handle, and the other end of the water pipe is fixedly connected to the water tank. I believe that the structure of the water pipe with six turns has been pre-treated and stabilized before leaving the factory. When using it, people usually hold the toothbrush handle and stretch the water pipe to aim the toothbrush at the user's teeth. During the stretching process, the water pipe is like a stretched spring (spiral), and the hand holding the handle will feel the increasing elastic restoring force of the water pipe to return to the original rotation position.

Figure 2. The pulled open water pipe

Since the position of the water flosser is not far from the user's teeth, when it is used, the six ring structures will be deformed but still retained (as shown in Figure 2), and people do not need to, and will not, quickly pull the water pipe into a straight pipe without the ring structure. Common sense shows that if the water pipe is straightened in an instant like this, the water pipe material will be damaged or broken.

Let’s study another similar example:

Figure 3. The coiled charging cable

Figure 3 shows the charging cable used by the author when charging his electric car at home. One end of the cable is a plug (connected to the charging port on the car), which is a nearly cylindrical shape with a vehicle plug lock and a release button. The other end is a rectangular wire status indicator, which is connected to a 120-volt plug . For convenience, the cable is usually coiled into 8 turns and tied with a strap and placed in the lower layer of the trunk of the car. Due to being placed like this for a long time, the cable has adapted to this 8-turn coil structure and will not automatically return to a straight shape even if the strap is loosened. When charging at home, the author needs to untie the fixed binding strap, first insert the cylindrical plug into the charging port on the car, then hold the rectangular indicator at the other end to stretch the charging cable and insert the power plug into the power supply. At this time, like the stretching of the PVC water pipe in the above example, during the process of pulling the cable apart, the cable still retains the original 8-turn structure (see Figure 4), and the cable forms an increasingly strong elastic recovery force on the plug parts at both ends.

Figure 4. The cable being pulled apart.

Fortunately, the power supply in the yard is not far from the electric car, so there is no need to forcibly straighten the cable. Common sense tells me that if the cable is straightened quickly and forcibly, the cable system will inevitably suffer structural damage.

There are countless examples of this in life.

Such examples all involve long "wires" or long "thin tubes" made of a given length and a certain elastic and plastic material (such as plastic, carbon fiber, rubber, steel or plant fiber, etc.). Assume that before use, the "wire" (or long "thin tube") is coiled into a spiral shape with n (integer) turns around a thicker (can be virtual) axis . When in use, fix one end of the spiral and hold the other end of the spiral (that is, the spiral is prohibited from twisting at this end point) to pull and stretch the spiral. In this process, the shape of the spiral will change, but its structure of n turns will be maintained and will not be instantly pulled into a straight line. Unless the internal structure of the curved material undergoes a qualitative change, the elasticity of the material completely disappears and becomes a pure plastic material, or breaks, the topological structure of the n turns will change. This is undoubtedly related to profound physical and chemical mechanisms.

Therefore, from a physical perspective, it makes sense to regard the number of rotations n of the above wire as a topological invariant.

In mathematics, such examples involve the problem of space curve invariants in differential geometry.

The first thing to explain is why a thin line or long tube made of material can be represented by a space curve?

Although a space curve is a one-dimensional smooth curve without thickness in space, within the scope of differential geometry, it is generally required that the expression describing the one-dimensional parameter t on which the position vector of each point on the curve depends must have continuous third-order derivatives of the parameter. On this basis, a three-dimensional coordinate frame (also called Frenet coordinate frame) composed of unit tangent (line) vector T , unit principal normal (line) vector N and unit binormal (line) vector B can be established along each point of the curve (Figure 5).

Figure 5.  Tangent vector  T , normal vector  N  and binormal vector  B of a space curve.

Photo from English Wikipedia : Salix alba

Illustration of the frenet frame. Created using Singsurf.org and Javaview.de postprocessed with gimp.

The strict definitions and mathematical expressions of the vectors T , N , and B will be introduced in the second section of this article, and the reader can also find them in any differential geometry textbook ([1],[2],[3]).

As the parameters change continuously, the coordinate frame attached to the curve points also moves together and rotates around the tangent vector. Since it is a space curve, this rotation must exist almost everywhere. Therefore, it can be imagined that the entire space curve is wrapped in a thin tube (without thickness) with a radius of unit length, so that when the coordinate frame changes continuously with the points on the curve, the end arrows of the unit principal normal vector and the binormal vector leave their own tracks on the wall of the thin tube. Since it is generally assumed that the space curve under study does not have any intersection points, it can be considered that the above-mentioned imagined thin tube wrapped outside the curve does not touch each other. This is because the length unit can be arbitrarily selected to be appropriately small to avoid the possibility of thin tubes crossing. Therefore, the mathematical space curve together with the imaginary thin tube can represent a thin line or long tube with a fixed thickness made of actual material.

Mathematically, the n -turn curve in the above example is equivalent to an n- turn cylindrical spiral in the sense of differential topology . Figure 6 shows a cylindrical spiral with 10 turns:

Figure 6. A cylindrical helical segment with 10 turns

The number of rotations n of a curve segment belongs to the intrinsic large-scale (also called global) properties of the curve. To more fully study the global properties of topological manifolds, topology generally uses closed manifolds. A closed manifold is a compact manifold without a boundary ( see Wikipedia https://en.wikipedia.org/wiki/Closed_manifold ). This is because there are some special problems that need to be handled when calculating differentials at the boundary, such as the concept of one-sided derivatives. The above cylindrical spiral segment is not a one-dimensional closed manifold (because it has two boundary endpoints that are not one-dimensional open sets), so it is not ideal.

Just because the unit tangent vectors at the two endpoints of the spiral segment have the same directions as the unit principal normal vector and the unit binormal vector, we can imagine that through topological transformation, the two ends of the spiral can be regarded as the same point (as shown in Figure 7), making it  a closed spatial curve with no boundaries and n  loops (one-dimensional closed manifold).

Figure 7. A closed space curve with 10 loops

To illustrate the rationality of the above (differential) topological approach, let us study the construction method of the Klein bottle (Figure 8), a famous example in the study of two-dimensional closed manifolds.

Figure 8. Klein bottle (from Wikipedia, image by  Tttrung  )

According to Wikipedia's explanation of the entry "Klein bottle": first imagine a red wine bottle with a hollow bottom. Mathematically, it is imagined as a two-dimensional oriented surface with two circular one-dimensional boundaries. One boundary is the bottle mouth, and the other is the bottom line of the remaining bottle surface after planning the bottom of the bottle. Then the neck is extended, twisted outward and extended into the interior of the bottle, and the bottle mouth is connected to the bottom boundary. In this way, the Klein bottle is virtually constructed.

Comparing the above method of constructing n closed curves from n cylindrical spiral segments with the construction method of the Klein bottle, it is not difficult to see that the former is feasible in three-dimensional space, while the latter is not. Therefore, the former construction method appears to be more realistic and reasonable.

Intuitively, the number of rotations (circles) n of the above space closed curve should be a differential topological invariant. If we remove the word "differential" and only look at it from a topological perspective, the space closed curve shown in Figure 7 can be continuously transformed (or homotopically transformed) into a plane circle. In this case, the number of rotations cannot be regarded as a topological invariant, so it is not so simple to define this invariant mathematically.

It can be found that only by properly combining the mathematical properties of the curve with its physical connotation, or by "materializing" the structure related to the space curve from the perspective of differential topology, can a reasonable definition be given. This part will be introduced in the second section of the blog post.

(to be cotinoued)