*33*

fix one end, twist the rope at the other end in the direction of the arrow, and the middle of the rope will be automatically screwed together (my poor charging line). This is because the internal stress is generated when the rope is twisted. The twisting coil in the middle can reduce the torsion, that is, reduce the stress…

well, I won’t talk about physics, but about mathematics

in fact, this is because it is the number of circles, the number of twists and turns, but the same trace invariant, so the sum of the number of twists and turns is conserved

if the number of twists is reduced, the number of twists will increase accordingly, and vice versa. American geometer white proved the formula in his doctoral thesis published in 1969

OK, next I’ll try to explain

first, the first concept. If a knot or link can continuously become another without cutting or bonding, then the two knots or links are the same mark, such as two knots with the same mark:

two links with the same mark:

what is the difference between a knot and a link? In short, a knot is a circle, and a chain link is any number of circles. Therefore, knot is also a kind of link, which is called ordinary link

the essence of the same trace can be characterized by three basic transformations (commonly referred to as elementary transformations):

R1 (eliminate or add a volume):

R2 (eliminate and add an overlapping “polygon”):

R3 (triangular transformation):

note that these three elementary transformations are carried out locally, and no other lines can intervene in the transformed part, For example:

correct transformation will get different results:

how to prove that two links have the same mark? We can find a way to change one into another through elementary transformation. For example, the upper left and lower left knots in the figure below have the same mark:

but how to prove that the two links have different marks? This is difficult, because the inability to find a method of elementary transformation does not mean that the method must not exist. Therefore, we need to use some other tools, such as Jones inequality. However, this has little to do with this article, so I won’t introduce it in detail

each closed curve has two opposite detour directions. When we select the direction of each circle of the link, it becomes a directed link.

fix one end, twist the rope at the other end in the direction of the arrow, and the middle of the rope will be automatically screwed together (my poor charging cable). This is because the internal stress is generated when the rope is twisted. The twisting coil in the middle can reduce the torsion, that is, reduce the stress…

well, I won’t talk about physics, but about mathematics

this is actually because it is the number of circles, the number of twists and turns, and the same trace invariant, so the sum of the number of twists and turns is conserved

if the number of twists is reduced, the number of twists will increase accordingly, and vice versa. American geometer white proved the formula in his doctoral thesis published in 1969

OK, next I’ll try to explain

first, the first concept. If a knot or link can continuously become another without cutting or bonding, then the two knots or links are the same mark, such as two knots with the same mark:

two links with the same mark:

what is the difference between a knot and a link? In short, a knot is a circle, and a chain link is any number of circles. Therefore, knot is also a kind of link, which is called ordinary link

the essence of the same trace can be characterized by three basic transformations (commonly referred to as elementary transformations):

R1 (eliminate or add a volume):

R2 (eliminate and add a superimposed “polygon”):

R3 (triangular transformation):

note that these three elementary transformations are carried out locally, and no other lines can intervene in the transformed part, For example:

correct transformation will get different results:

how to prove that two links have the same mark? We can find a way to change one into another through elementary transformation. For example, the upper left and lower left knots in the figure below have the same mark:

but how to prove that the two links have different marks? This is difficult, because the inability to find a method of elementary transformation does not mean that the method must not exist. Therefore, we need to use some other tools, such as Jones inequality. However, this has little to do with this article, so I won’t introduce it in detail

each closed curve has two opposite detour directions. When we select the direction of each circle of the link, it becomes a directed link

next, we assign a sign to the intersection of the projection of the directed link:

that is, if the minimum rotation angle of the arrow on the line to the arrow on the line is counterclockwise, the intersection is positive; Clockwise is negative

the sum of the positive and negative signs of all intersections of the projection of a directed link is called the number of turns, which is recorded as. The twist number is obviously unchanged under the two elementary transformations of R2 and R3, but R1 will change it… So the twist number is not a same mark invariant

what is the same trace invariant? Number of circles. It measures the degree to which two directed closed curves surround each other

let it be two branches of a directed link, then the number of circles is defined as half of the sum of the sign of the intersection with. In this way, R1, R2 and R3 will not change the value of the surround number, so the surround number is the same trace invariant

note that the intersection when calculating the number of circles does not include the self intersection of, nor does it include the self intersection of, nor does it include the intersection with other branches. When there are only two branches, it can be abbreviated as

for example:

this can show that the two links in the figure above have different marks because of their different number of circles

the surround number was first proposed by Gauss when studying electromagnetic phenomena, but I won’t introduce it…

OK, what can the surround number do? Don’t worry, look at the picture first:

we call the shape on the left of the figure above twist and the shape on the right twist> there are two sides of the letter that can be twisted or not. Br>> please consider each other. The Mobius band is not considered because it is unilateral and has only one side

the two sides of the belt take the same direction, so you can get a directed link. Its two branches, the two sides, are denoted as

the contribution of the torsion part to the number of circles is:

the shape of the upper side is called a positive torsion cycle, and the shape of the lower side is called a negative torsion cycle. Represents the sum of the number of torsion cycles on the whole projection (positive and negative cancellation)

the contribution of the winding part to the number of turns is:

this is exactly equal to the contribution of this part to the number of turns

the number of circles is the sum of the contributions of the torsion part and the winding part, so

for example:

a conservation law can be seen from the

formula. When the belt deforms continuously in space, the twist number and twist number will change, but their sum is a same trace invariant

generally, a rope can be considered as a belt. One side of the belt is the center line of the rope, and the other side is a sign line on the surface (recording the torsion around the center line), as shown in the figure:

theref

ore, for the rope, the sum of the number of turns and the number of turns does not change. This is about (?) You can answer the main question

however, it is slightly different from what was said at the beginning. Is the average of the number of turns of the projection when viewed from all possible directions; There are more complex definitions, which will not be discussed in detail here

if you are interested, you can look at the Book Mathematics of rope circles. You only need to have the knowledge of high school mathematics to understand most of the contents of the book, and if you have a little knowledge of calculus, you can understand all the contents.