纽结理论是拓扑学的一个重要分支,虚拟纽结理论是经典纽结理论的推广,对它的研究是通过一种图解理论来展开的。虚拟纽结多项式是一类以多项式表达的虚拟纽结不变量,例如Arrow多项式和Wriggle多项式。Affine index多项式是以虚拟纽结图...纽结理论是拓扑学的一个重要分支,虚拟纽结理论是经典纽结理论的推广,对它的研究是通过一种图解理论来展开的。虚拟纽结多项式是一类以多项式表达的虚拟纽结不变量,例如Arrow多项式和Wriggle多项式。Affine index多项式是以虚拟纽结图的整数标记定义的单变量多项式。本文主要计算一类特殊虚拟纽结的Affine index多项式。按照Cheng着色的规则,对虚拟纽结图的每一段弧进行整数标记,计算每个经典交叉点的指标值,进而得到这类特殊虚拟纽结的Affine index多项式的表达式。Knot theory is an important branch of topology. Virtual knot theory is a generalization of classical knot theory, and its research is carried out through a graphic theory. The virtual knot polynomial refers to a class of virtual knot invariant expressed by polynomials, such as the Arrow polynomial and the Wriggle polynomial. The affine index polynomial is a univariate polynomial defined by the integer label of a virtual knot graph. In this paper, we mainly calculate affine index polynomials for a special class of virtual knots. According to the rules of Cheng coloring, we will integer label each arc of the virtual knot graph and calculate the index value of each classical crossings, and then get the expression of the affine index polynomial of this special virtual knot.展开更多
虚拟纽结理论中主要研究对象为纽结和链环。虚拟纽结多项式是一类以多项式表达的虚拟纽结不变量,例如零多项式和Writhe多项式。文章通过研究零多项式和虚拟链环的writhe多项式给出了一个新的多项式。这个多项式通过对虚拟链环的计算得...虚拟纽结理论中主要研究对象为纽结和链环。虚拟纽结多项式是一类以多项式表达的虚拟纽结不变量,例如零多项式和Writhe多项式。文章通过研究零多项式和虚拟链环的writhe多项式给出了一个新的多项式。这个多项式通过对虚拟链环的计算得出原始虚拟纽结的多项式,证明了这个多项式是一虚拟纽结不变量,给出了其性质及一实例计算。The main research objects of virtual knot theory are knot and link. Virtual knot polynomials are a class of virtual knot invariants expressed as polynomials, such as the zero polynomial and the writhe polynomial. In this paper, a new polynomial is given by studying zero polynomials and writhe polynomials of virtual links. The polynomial of the original virtual knot is obtained by computing the virtual link. It proves that the polynomial is a virtual knot invariant, and properties and computational examples are given.展开更多
纽结理论中主要研究对象为纽结和链环,对它们的研究是通过所对应的纽结投影图来展开的。纽结多项式是一类以多项式表达的纽结不变量,例如Alexander多项式和Jones多项式。HOMFLY多项式是一个双变量的Laurent多项式,两个变量分别是m和l,是...纽结理论中主要研究对象为纽结和链环,对它们的研究是通过所对应的纽结投影图来展开的。纽结多项式是一类以多项式表达的纽结不变量,例如Alexander多项式和Jones多项式。HOMFLY多项式是一个双变量的Laurent多项式,两个变量分别是m和l,是继Jones多项式之后,又一计算纽结不变量的多项式。Brunnian链环是一类特殊的链环,拆去任何单个分量都会生成一个平凡链环。本文主要结合HOMFLY多项式的定义和性质,应用拆接关系研究计算Brunnian链环的HOMFLY多项式。The main research objects in knot theory are knots and links, but the research on them is carried out through the corresponding knot projection diagram. The knot polynomial refers to a class of knot invariant expressed by polynomials, such as the Alexander polynomial and the Jones polynomial. The HOMFLY polynomial is a bivariate Laurent polynomial with two variables m and l, which is another important polynomial for calculating knots after the Jones polynomial. Brunnian link is a special class of link in which the complement of any one component is a trivial link. In this paper, we state and calculate the HOMFLY polynomials of Brunnian link by using the definition and properties of HOMFLY polynomials and the disconnection relation.展开更多
It is proved that the coefficients of conway polynomial in z0,z1 and z2 and all ambient isotopic invariants. In particular,the coefficient in z2 of conway polynomial for link L with two components is only depend on sw...It is proved that the coefficients of conway polynomial in z0,z1 and z2 and all ambient isotopic invariants. In particular,the coefficient in z2 of conway polynomial for link L with two components is only depend on switch crossings in pairs.展开更多
文摘纽结理论是拓扑学的一个重要分支,虚拟纽结理论是经典纽结理论的推广,对它的研究是通过一种图解理论来展开的。虚拟纽结多项式是一类以多项式表达的虚拟纽结不变量,例如Arrow多项式和Wriggle多项式。Affine index多项式是以虚拟纽结图的整数标记定义的单变量多项式。本文主要计算一类特殊虚拟纽结的Affine index多项式。按照Cheng着色的规则,对虚拟纽结图的每一段弧进行整数标记,计算每个经典交叉点的指标值,进而得到这类特殊虚拟纽结的Affine index多项式的表达式。Knot theory is an important branch of topology. Virtual knot theory is a generalization of classical knot theory, and its research is carried out through a graphic theory. The virtual knot polynomial refers to a class of virtual knot invariant expressed by polynomials, such as the Arrow polynomial and the Wriggle polynomial. The affine index polynomial is a univariate polynomial defined by the integer label of a virtual knot graph. In this paper, we mainly calculate affine index polynomials for a special class of virtual knots. According to the rules of Cheng coloring, we will integer label each arc of the virtual knot graph and calculate the index value of each classical crossings, and then get the expression of the affine index polynomial of this special virtual knot.
文摘虚拟纽结理论中主要研究对象为纽结和链环。虚拟纽结多项式是一类以多项式表达的虚拟纽结不变量,例如零多项式和Writhe多项式。文章通过研究零多项式和虚拟链环的writhe多项式给出了一个新的多项式。这个多项式通过对虚拟链环的计算得出原始虚拟纽结的多项式,证明了这个多项式是一虚拟纽结不变量,给出了其性质及一实例计算。The main research objects of virtual knot theory are knot and link. Virtual knot polynomials are a class of virtual knot invariants expressed as polynomials, such as the zero polynomial and the writhe polynomial. In this paper, a new polynomial is given by studying zero polynomials and writhe polynomials of virtual links. The polynomial of the original virtual knot is obtained by computing the virtual link. It proves that the polynomial is a virtual knot invariant, and properties and computational examples are given.
文摘纽结理论中主要研究对象为纽结和链环,对它们的研究是通过所对应的纽结投影图来展开的。纽结多项式是一类以多项式表达的纽结不变量,例如Alexander多项式和Jones多项式。HOMFLY多项式是一个双变量的Laurent多项式,两个变量分别是m和l,是继Jones多项式之后,又一计算纽结不变量的多项式。Brunnian链环是一类特殊的链环,拆去任何单个分量都会生成一个平凡链环。本文主要结合HOMFLY多项式的定义和性质,应用拆接关系研究计算Brunnian链环的HOMFLY多项式。The main research objects in knot theory are knots and links, but the research on them is carried out through the corresponding knot projection diagram. The knot polynomial refers to a class of knot invariant expressed by polynomials, such as the Alexander polynomial and the Jones polynomial. The HOMFLY polynomial is a bivariate Laurent polynomial with two variables m and l, which is another important polynomial for calculating knots after the Jones polynomial. Brunnian link is a special class of link in which the complement of any one component is a trivial link. In this paper, we state and calculate the HOMFLY polynomials of Brunnian link by using the definition and properties of HOMFLY polynomials and the disconnection relation.
文摘It is proved that the coefficients of conway polynomial in z0,z1 and z2 and all ambient isotopic invariants. In particular,the coefficient in z2 of conway polynomial for link L with two components is only depend on switch crossings in pairs.
