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

Abstract

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 <i>f</i>(<I>L</I>), and to prove <i>f</i>(<I>L</I>) 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 <i>f_k</i>(<I>L</I>), the property of <i>f</i>(<I>L</I>) is obtained by using the properties of <i>f_k</i>(<I>L</I>). 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.