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The number of faces, edges and vertices of the introduction is not independent, but is linked in a simple way. It uses the earliest examples of topological invariants to distinguish solids with different topologies. Topology, one of the most important and powerful fields in pure mathematics, is the study of the invariant properties of geometric objects after continuous deformation. It helps us understand how enzymes act on DNA in cells and why the motion of celestial bodies can be chaotic.

Euler cube as the 19th century drew to a close, mathematicians began to develop a new geometry in which familiar concepts such as length and angle were no longer the key, and there was no difference between triangles, squares and circles. It was originally called location analysis, but mathematicians soon found another name: topology.

Descartes noticed the topology when he thought about the five regular polyhedrons of Euclid in 1639. Descartes therefore turned his attention to the positive cube, and it was at this time that he noticed the law of numbers about the positive cube. A cube has 6 faces, 12 sides, and 8 vertices:

A dodecahedron has 12 faces, 30 sides, and 20 vertices:

An icosahedron has 20 faces, 30 sides and 12 vertices; the sum of 20-30 + 12 equals 2. The same relationship applies to tetrahedrons and octahedrons. In fact, it applies to solids of any shape, regular or irregular. If a solid has F faces, E edges, and V vertices, then:

Descartes believes that this formula is only a small discovery and has not been published. It was not until a long time later that mathematicians saw this simple equation as the first step towards topology. In the 19th century, the three pillars of pure mathematics were algebra, analysis and geometry. By the end of the 20th century, it had become algebra, analysis and topology. Topology is often described as "plasticine geometry" in which lines can be bent, contracted or stretched, while circles can be squeezed to become triangles or squares, and it is important to maintain continuity. Continuity is not only a basic aspect of the natural world, but also a basic feature of mathematics. Today, we mainly use topologies indirectly. Some properties of quantum field theory and iconic molecule DNA need to be understood by topology.

Euler proved and published this relationship in 1750 and 1751. The expression of F-E + V looks quite casual, but it has a very interesting structure. The face (F) is a two-dimensional polygon; the edge (E) is a line and is one-dimensional; the vertex (V) is a point and is 0-dimensional. In the expression + F-E+V, "+" represents an even dimension and "-" represents an odd dimension. This means that solids can be simplified by merging faces or deleting edges and vertices, and these changes do not change the results of F-E + V.

Now, let me explain. As shown in the figure:

Key steps to simplify solids. From left to right: (1) start; (2) merge adjacent faces; (3) retain the "tree" after all the faces are merged; (4) remove an edge and a vertex from the tree; (5) end, first, turn the solid into a ball, its edge is the curve on the ball. If two faces share the same edge, then you can delete the edge and merge the two faces into one. Because this merger reduces both F and E by 1, it does not change the result of F-E + V. Keep doing this until you get a face that covers almost the entire sphere (except for this face, leaving only the lower edges and vertices). They must form a network without a closed loop, because any closed loop on a sphere is separated by at least two faces: one inside the closed loop and the other outside the closed loop.

This process continues until only one vertex is left on a sphere without any features. Now V = 1, E = 0, F = 1. F-E + V = 1-0 + 1 = 2. But since F-E + V is constant at each step, its initial value must also be 2, which is what we want to prove.

This proof has two components. One is to simplify the process: delete a face and an adjacent edge, or delete a vertex and an edge that intersects it. The other is the invariant, that is, the mathematical expression that remains unchanged whenever a step in the simplification process is performed. As long as these two components exist at the same time, you can simplify the value of the invariant of any initial object as much as possible, and then calculate the value of the invariant of this simplified version. Because it is an invariant, the two values must be equal. Because the end result is simple, invariants are easy to calculate.

In fact, Descartes' formula does not apply to any solid. The most common unsuitable solid is the picture frame. Imagine a four-sided photo frame made of wood, each with a rectangular cross section connected by a 45 °bevel at the four corners, as shown in the following illustration. The wood on each side contributes 4 faces, so F = 16. Each piece of wood also contributes 4 edges, but the miter creates 4 edges on each corner, so E = 32. Each corner contains 4 vertices, so V = 16. So F-E + V = 0.

What's the problem?

Left: Fmure + V = 0 photo frame. Right: there is no problem with the F-E + V invariance of the final structure after smoothing and simplifying the photo frame. There is no problem with simplifying the process. But if you deal with the frame, always removing a face on one side, or a vertex on one side, then the final simplified configuration is not a single vertex on a single face. As shown on the right of the above figure: F = 1, V = 1, E = 2. At this stage, removing an edge only merges the only remaining face with itself, so the change to the number is no longer offset. That's why we stopped, but we still got the answer: for this configuration, F-E + V = 0. Therefore, the method is executed perfectly. It just produces different results for the picture frame. There must be some basic differences between a picture frame and a cube, which are reflected by the invariant F-E + V.

Earlier, I told you to "change a solid into a ball". But it's impossible for a picture frame. Even if it is simplified, it does not look like a sphere. It is a torus that looks like a tire with a hole in the middle. However, Fmure + V remains the same. This proof tells us that any solid that can be deformed into a torus satisfies a slightly different equation: F-E + V = 0. Therefore, we have the basis for strictly proving that the torus cannot be transformed into a sphere, that is to say, the two surfaces are different in topology.

Of course, this is intuitively obvious, but now we can use logic to support intuition. Just as Euclid started from the obvious properties of points and lines and formalized them into strict geometric theories, mathematicians in the 19th and 20th centuries developed strict formal topological theories.

Left: 2-hole torus right: 3-hole torus A solid such as a torus with two or more holes, as shown in the figure above. The results show that any solid deformable to a 2-hole torus satisfies F-E + V =-2, and any solid deformable to a 3-hole torus satisfies F-E + V =-4. Generally speaking, any solid deformable to a g-hole torus satisfies F-E + V = 2-2g.

Following the ideas of Descartes and Euler, we find the relationship between the quantitative properties of solids (the number of faces, vertices and edges) and the properties with holes. We call F-E + V the Euler characteristic number of the cube.

We calculate the number of holes, which is a quantitative operation, but the "hole" itself is qualitative because it is not a solid feature at all. Intuitively, it is a region in space and a solid is not. In fact, the more you start to think about what a hole is, the more you will realize that defining a hole is tricky, as shown below:

This is my favorite example. It's called "the hole in the hole". Obviously you can put one hole through another.

The situation is getting more and more complicated. By the end of the 19th century, they were everywhere in mathematics-in complex analysis, algebraic geometry and Riemannian differential geometry. To make matters worse, high-dimensional solid analogues occupy a central position in all fields of pure and applied mathematics. The dynamics of the solar system requires each object to have six dimensions. They have hole analogues in higher dimensions. In any case, it is necessary to bring some order to this new field. The answer is: invariant.

The idea of topological invariants can be traced back to Gauss's study of magnetism. He is interested in how magnetic lines of force and power lines are connected to each other. He defines the number of connections, that is, the number of times one magnetic line of force revolves around another. This is a topological invariant: if the curve deforms continuously, it remains the same. Gauss's student John Liszt and Gauss's assistant Auguste Mobius learned more about Gauss's research for the first time. Liszt introduced the word "topology" in the study of Topology in 1847, while Mobius defined the role of continuous deformation.

Liszt is looking for a generalization of Euler's formula. The expression FE + V is a combinatorial invariant. The number of holes g is a topological invariant: no matter how the solid deforms, it will not change as long as the deformation is continuous. Topological invariants capture the qualitative conceptual characteristics of shapes, and combinatorial functions provide a calculation method. The combination of the two is very powerful because we can use conceptual invariants to consider shapes and combinatorial invariants to determine what we want to discuss.

In fact, this formula completely avoids the thorny problem of defining a "hole". Instead, we define the "number of holes" as a package that neither defines holes nor calculates how many holes there are. How do you do it exactly? Is to rewrite the Euler formula F-E + V = 2-2g into this form:

Now we can calculate g by "picture" on the solid, calculate E and V, and then put these values into the formula. Because the expression is an invariant, no matter how we split the entity, we always get the same answer. But everything we do does not depend on the definition of the hole. On the contrary, the "hole number" has become an intuitive explanation.

This is a major breakthrough in one of the core questions of topology: when can one shape change continuously into another? In other words, as far as topologists are concerned, are the two shapes the same? If they are the same, their invariants must be the same; conversely, if the invariants are different, the shapes will be different. Because the sphere has the Euler characteristic number 2 and the torus has the Euler characteristic number 0, it is impossible to continuously deform the sphere into a torus.

Less obviously, the Euler indicative number shows that this inexplicable "hole in the hole" is actually a camouflaged three-hole torus. Most of the complexity of the surface does not come from the inherent topology of the surface, but from the way I choose to embed it in space.

The first really important theorem in topology comes from the formula of Euler's indicative number. It is a complete classification of surfaces, the two-dimensional shape of a surface, such as a sphere or torus. In addition, some technical conditions have been imposed: the surface should have no boundaries, and the scope should be limited (the term is "compact").

For this purpose, the surface is essentially described; that is, it does not exist in the surrounding space. One way is to think of the surface as a number of polygonal regions that stick together along the edges according to specific rules.

The possibility of gluing the edges of a square together to form a torus bonding boundary leads to a rather strange phenomenon: a surface with only one side. The most famous example is the Mobius belt, which is a rectangular band whose ends are glued together in a 180 °rotation. The Mobius band has only one side because the two separate edges of the rectangle are connected by semi-twisting.

We can easily make a Mobius band because it can be naturally embedded in three-dimensional space. This band has only one side, that is, if you start to draw one surface of it, and then continue to draw, you will eventually cover the whole surface, front and back.

This is because the semi-torsion connects the front and back. This is not an inherent description because it depends on embedding bands into space, and there is an equivalent, more professional feature called orientability, which is inherent.

If we glue the two edges of a rectangle together, like a Mobius belt, and then glue the other two edges together, there is no need for any distortion. The surface is depicted as a cross that looks like the neck of a bottle poking through the side wall and connected to the bottom. It was invented by Klein and is called the Klein bottle.

The Klein bottle has no boundaries and is compact, so any surface classification must include it. It is the most famous of all families of one-sided surfaces.

In many areas of mathematics, surfaces appear naturally. They are very important in complex analysis. in complex analysis, surfaces are related to singularities on which the function behaves abnormally-for example, derivatives do not exist. Singularity is the key to many problems in complex analysis. Because the singularity is related to the surface, the topological structure of the surface provides an important technique for complex variable analysis.

Most modern topologies are highly abstract, and many topologies occur in four-dimensional or multi-dimensional spaces. We can have a sense of the subject in a more familiar environment: kink. In the real world, a knot is made with a rope. Topologists need a way to prevent the knot from leaving the end of the knot, so they connect the ends of the knot together to form a closed loop. A knot is a circle embedded in space. In essence, the topological structure of the knot is the same as that of the circle, but in this case, the important thing is the position of the circle in the surrounding space. This seems to run counter to the spirit of topology, but the essence of the knot lies in the relationship between the string ring and the space around it. By considering not only the loop, but also its relationship with space, topology can solve important problems about nodes. These include:

How do we know a knot is really tied?

How do we distinguish between topologically different knots? In other words, whether two knots can change from one smooth terrain to another without having to cut the knot itself is still considered a complex mathematical problem. The knot invariant is a powerful tool to help answer this question, which we will introduce next.

Can we classify all possible knots?

It took the Scottish theoretical physicist Peter Tait years to develop the earliest knot classification table. In 1910, Mark Stern introduced the concept of knot group. In 1928 James Ward Alexander introduced the knot polynomial, a more manageable invariant. These are all important advances in the development of knot theory.

After about 1960, the conclusion entered the low ebb of topology, waiting for creative insights. In 1984, New Zealand mathematician Vaughn Jones invented a new knot invariant, called Jones polynomial, which is also defined by knot graphs and three types of movement. However, these movements do not retain the topology type of the knot. However, surprisingly, this idea is still feasible, and the Jones polynomial is a knot invariant.

Jones's discovery won him the Fields Prize. It also led to the outbreak of new invariants. In 1985, four different groups of mathematicians (8 individuals) simultaneously discovered the same generalization of Jones polynomials and submitted their papers to the same magazine. The four proofs were all different, and the editor persuaded the eight authors to publish a joint article. Their invariants are often referred to as HOMFLY polynomials (based on the initials of names). But even Jones polynomials and HOMFLY polynomials do not fully answer the three questions of knot theory. A systematic classification of all possible knots is still a pipe dream for mathematicians.

Topologies have many uses, but they are usually indirect. For example, our understanding of chaos is based on the topological characteristics of dynamical systems.

More esoteric applications of topology appear at the forefront of basic physics. Here, the main "consumer" of topology is quantum field theorists, because superstring theory, the unified theory of quantum mechanics and relativity, is based on topology. Here, a similar Jones polynomial appears in the context of the Feynman diagram in the junction theory, which shows how quantum particles, such as electrons and photons, move, collide, merge and split through space-time. Feynman diagram is a bit like a knot picture.

For me, one of the most attractive applications of topology is its increasing use in biology to help us understand how life molecules DNA works. Because DNA is a double helix structure, like two spiral stairs intertwined with each other. These two chains are intricately intertwined, and this complex topology must be taken into account in important biological processes, especially the way DNA is replicated during cell division.

Some enzymes, called recombinant enzymes, cut off two strands of DNA and then reconnect in different ways. To determine the role of the enzyme in cells, biologists applied the enzyme to the closed loop of DNA. Then they used an electron microscope to observe the shape of the modified ring. If the enzyme connects different chains together, the image is a knot:

If the enzyme separates these chains, the image shows two connected rings. Methods of knot theory, such as Jones polynomials and another theory called "entanglement", make it possible to study the occurrence of knots and connections, which provides detailed information about the action of enzymes.

In general, you won't encounter topologies in your daily life. But behind the scenes, topology runs through mainstream mathematics, enabling the development of other technologies with more obvious practical uses. This is why mathematicians think that topology is very important, while people outside mathematics have hardly heard of it.

This article comes from the official account of Wechat: Lao Hu Shuo Science (ID:LaohuSci). Author: I am Lao Hu.

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