A Knot Invariant Defined Based on the Skein Relation with Two Equations

: Knot theory is a branch of the geometric topology, the core question of knot theory is to explore the equivalence classification of knots; In other words, for a knot, how to determine whether the knot is an unknot; giving two knots, how to determine whether the two knots are equivalent. To prove that two knots are equivalent, it is necessary to turn one knot into another through the same mark transformation, but to show that two knots are unequal, the problem is not as simple as people think. We cannot say that they are unequal because we can't see the deformation between them. For the equivalence classification problem of knots, we mainly find equivalent invariants between knots. Currently, scholars have also defined multiple knot invariants, but they also have certain limitations, and even more difficult to understand. In this paper, based on existing theoretical results, we define a knot invariant through the skein relation with two equations. To prove this knot invariant, we define a function f ( L ), and to prove f ( L ) to be a homology invariant of a non-directed link, we need to show that it remains constant under the Reideminster moves. This article first defines the f k ( L ), the property of f ( L ) is obtained by using the properties of f k ( L ). In the process of proof, the induction method has been used many times. The proof process is somewhat complicated, but it is easier to understand. And the common knot invariant is defined by one equation, which defining the knot invariant with two equations in this paper.


Introduction
Topology is a branch of mathematics that is mainly studying the properties that objects remain unchanged after continuous transformations. And a more attractive area of topology is knot theory, there are two main reasons: One is because it studies real geometric phenomena in real life; the other is because it is mysterious to use different methods to study it thoroughly. Because of this, knot theory meets topology like differential geometry, number theory, algebraic geometry, matrix theory and group theory. Long long ago, people have already tie with rope, the most famous is: knot rope can remember, that is to record things in life by tying knots on the rope. knots can be seen everywhere in people's daily life, such as the popular Chinese knot during the Spring Festival in recent years, tying things with rope, sewing clothes, tying shoelaces, knitting sweaters and so on. On different occasions, people use different knots, so how to describe the knot mathematically? Although the famous allusion to the knot rope appeared before the invention, but from the mathematical perspective began in the 19th century, started by the German mathematician Calfried Gauss, he studied the nature of the electromagnetic field, found that the circle number between closed curves is related to the knot, this discovery laid a solid foundation for the study of knot theory, so the circle number became one of the main tools for scholars to study the knot. In 1867, Lord Kelvin regarded the atoms as the knot of the Etheric vortex, and one could classify the atoms with the aid of the classification of them. At the time, the hypothesis attracted many mathematicians, chemists, physicists to study knots, and knot theory emerged. A knot is the way a ring is embedded in 3 D space. Since any two knots are identical in the sense of homoderm, and we mainly study the way the curve is embedded in S 3 , we do not consider the length, the hardness, the degree of bending and the thickness of the curve itself. The knot theory is mainly to explore the equivalence classification of knots [1]. Trefoil knots are the simplest extraordinary knots. To prove that two knots are equivalent, it is necessary to turn one knot into another through the same mark transformation. But to show that two knots are unequal, the problem is not as simple as people think. We cannot say that they are unequal because we can't see the deformation between them. For the equivalence classification problem of knots, we mainly find equivalent invariants between knots. In 1928, the Alexander of the United States discovered the first knot polynomial invariant in history, in 1969; British Conway modified Alexander polynomials to obtain the Conway polynomial [2]. In 1984, the New Zealand mathematician Jones obtained Jones polynomials when studying operator polynombras, Subsequently, many scholars have found some more general knot invariants, such as chain ring branch number in [3], A family of polynomial invariants for flat virtual knots in [4], Whitney towers and abelian invariants of knots in [5], An infinite-rank summand of knots with trivial Alexander polynomial in [6], A polynomial time knot polynomial in [7], Regional knot invariants in [8], A free-group valued invariant of free knots in [9], An invariant for colored bonded knots in [10], Multi-switches and virtual knot invariants in [11], Invariants of knot diagrams in [12], which provide a strong theoretical basis for distinguishing knots, but these invariants are still insufficient, which prompted people to continue to find new knot invariants, this paper construct a new one to provide a new method for the classification of knots.

Preliminary Data
The knot on the left in Figure 1 is the simplest knot, commonly known as an ordinary or unknot; the knot on the right is called a trefoil knot. Figure 2 are several examples of additional knots.  A link is composed of finite knots in space, and they disintersect each other. If a link is composed of n nodes, it is called the link with n branches, and each knot forming the link is called a branch of the link. Obviously, the knot is a special link, and it has only one branch.
Obviously, in the plane, a link composed of finite multiple disjoint junctions must be a trivial link.
Below are several simple links:  Definition 2.2 [13] We say that two links 1 L and 2 L in 3 S are isotopic, written 1 2 L L ≈ , if there exists an isotopy 3 3 : Definition 2.3 [14] If a projection diagram of the link K meets the following conditions: (1) Only a finite points on the projection plot are overlapping points; (2) This finite overlap points are the cross sections of the two arcs on the link; (3) At each key, the image of the arc located at the knot above indicates a solid line, and the line disconnected at the secondary key indicates the image of the arc at the knot below; It is called a regular projection graph of the link K. Theorem 2.1 [15] Two links diagrams are equivalent if and only if there are finite multiple Reidemeister moves, make one turn into the other.
The three Reidemeister moves are performed as follows: (R1 is to eliminate or add a volume) (R2 is to add or eliminate a stacked binary) (R3 is a triangle change) Definition 2.4 Let K be a knot, and if K has a projection graph on which a n intersection transformation yields an ordinary knot, and no junction transformation for any projection graph doing less than n times of K, the number of solution knots of K is called n. written u (K)=n. Proposition 2.1 Any projection graph G, given a knot K so that its number of in tersections is that n, can always pass an intersection transformation of no more than n times and turn it into a projection graph of ordinary nodes.  Definition 3.1 [16] If an algebra is generated by , , , ', ', ' a b c a b c , 1, 2, n t n = ⋯ , and the following three conditions are met:

Proposal of Polynomial Invariants
( (3) ( ) Then this algebra generates a Z-algebra, written Σ . Theorem 2.1 Given an algebra Σ , there are homomorphic invariants with unique non-orientations of the links ( ) f L ∈ Σ , and there are the following two types of establishment: (1) ( ) n n f T t = (initial condition) (T n is the trivial link with n branches).
(2) When the intersection comes from the same branch,

Theorem Proving
Since ( ) k f L is uniquely determined by the above several properties, and each knot projection graph can decompose into a linear combination of a series of descent graphs, so a function ( ) Due to the ( ) k f L being met: So n U is isotopy to n T .
For ( ) The other points of the following certificate also meet the relationship formula: To facilitate writing, p L + indicates cross points p is L + ; pq L ++ indicates cross points p is L + , q is L + ; pq L +− indicates cross points p is L + , q is L − and so on. We will represent the considered intersections with the p. We think about the case ( ) ( ) such that ( ) , 1 If q is the first bad intersection on P L + . I Assume that q=p, the conclusion was established II Assume q p ≠ , if q is L + and the intersection is on the Judging from the above induction and the main induction method: And by algebras Σ , the conditions (1)(2) are easy to launch ' a c ac Therefore, it is available using the above formula: ( ) = , we can know that L is descending relative to b′.
if q is L + and the intersection q is derived from the same branch (Other cases may be similar).
The following certificate remains unchanged under the Reidemeister moves.
Let L be a link diagram that keeps the branching order unchanged, R is the Reidemeister moves on L If ( ) 1, ( ( )) 1 cr L k cr R L k ≤ + ≤ + , We use induction on ( ) b L : (a) If a bad intersection point p is not involved in the Reidemeister move, then: Suppose p is the type L + and the intersection is from the same branch, other cases are similar. It can be seen from the hypothesis: There is no other bad point except for the bad point involved in the Reidemeister moves. For the third Reidemeister move, if p is the intersection point from the same branch and p is L + .
For the first Reidemeister move, the basis point can always be transformed to make it good.
For the second Reidemeister move, there is only one case that cannot make the bad point better. That is, the intersections are from different branches and the intersection types are different. As shown in Figure 7. We use induction on ( ) b L : (1) ( ) b L =0, it is established by lemma 4.1.
(2) If established at ( ) b L t < (3) ( ) b L t = , Let p be a bad intersection.
If p is L + , ( ) b L t − < By induction: Since the disassembly relation is independent of orientation, and the polynomial invariant of each chain loop is finally represented as some linear combination of decreasing knot graphs, it is easy to know that this polynomial invariant is independent of orientation, that is, the invariant is the same trace invariant of the non-orientation link.
This completes the proof.

Conclusion
In this paper, knot invariants by defining a system of split equations containing two equations, while common knot invariants are defined by a split equation which is a new method. This paper demonstrates the newly defined polynomial invariants by adopting multiple induction; Although the proof process is complicated, it is easy to understand, Compared to its proof, its practical application is somewhat difficult, which requires us to study later.