Properties And Calculations Of Knot Invariants

Professor Yang Zhiqing^[20]constructed a new invariant with multiple skein relation in2018 and proved to be an extension of Homfly polynomial and Kauffman polynomial.Similarly,we combined the construction method of the kauffman bracket polynomial to construct a new group of skein relations,which defines a new formal invariant.The knot invariants with multiple skein relations we constructed is different from the traditional polynomial invariant:1)The first is that the invariant with multiple skein relations we defined is a formal knot invariant and is only defined for links with a number of branches less than or equal to 2;2)The second is we have introduced a number of new ways to smooth intersections at intersections.3)The third is to select different skein relations for any intersection of the projection graph according to whether the two arcs of the intersection are from the same branch;4)The fourth is the coefficients in the skein relations form a commutative ring,and there are non-trivial relations between the coefficients.This article presents a simplified version of the multi-skein relation invariant.For a given projection graph,using the Diamond Lemma,a unique polynomial value can be obtained.Finally,we calculate the polynomials of some common knots with defined formal knot invariants.Chapter 4 of this article studies everywhere 1-trivial knots.If crossing change is performed at any crossing,the knot diagram is changed into a knot diagram of the trivial knot,we say that such a knot diagram is everywhere 1-trivial.An everywhere 1-trivial knot is a knot with an everywhere 1-trivial knot diagram.We have proved several interesting conclusions for everywhere 1-trivial knots:1)If it is a positive knot,then it must be the trefoil knot.2)If an everywhere 1-trivial knot diagram is a positive diagram,then it is a trefoil knot minimal diagram or a trefoil knot diagram with four crossings.3)An almost positive knot can not be an everywhere 1-trivial knot.4)If a 2-almost positive knot is everywhere 1-trivial,then it must be the figure eight knot.Finally,similar to Tsukamoto^[30],we constructed a number of local transformations,and conjectured that any everywhere 1-trivial knot diagram can be transformed into the trefoil knot minimal diagram or the figure eight knot minimal diagram through those local transformations.