一类代数链环的棍棒指标估计

Estimation of stick index for a class of algebraic links

纽结和链环的合痕分类是三维流形理论研究的一个核心问题.纽结和链环的不变量是研究纽结和链环合痕分类的重要工具,寻找易于计算且具有较强分辨能力的纽结和链环的合痕不变量一直以来都是纽结和链环理论研究的一个热点问题.从一类代数链环对应的代数投影图出发,利用代数投影图与其加权树图的对应关系,通过投影图的初等R变换对应消去加权树图中无用的树桩和树枝,通过投影图的翻转变换对应调整加权树图中树桩的数目和位置.经过一系列投影图的初等R变换和翻转变换操作,将代数投影图及其对应加权树图进行标准化.借助于整数缠绕的分段线性构造、代数链环的标准代数投影图以及对应标准加权树图的关系,得到一类代数链环的棍棒指标估计.

The isotopy classification of knots and links is a core problem in the study of three-dimensional manifold theory.Invariants of knot and link are important tools for studying the classification of knots and links,it has been a hot topic in knot and link theory to search for knot and link invariants which are easy to calculate and have strong resolving power.It starts from the algebraic projection diagrams corresponding to a class of algebraic links,by using the corresponding relation between the algebraic projection diagram and its weighted tree graph,the unnecessary stumps and twigs in the weighted tree graph are eliminated through the elementary R operations of the projection diagram,the number and position of stumps in the weighted tree graph are adjusted by the flyping operations of the projection diagram.After a series of elementary R operations and flyping operations,the algebraic projection diagram of link and its corresponding weighted tree graph are standardized.With the help of the relation among the piecewise linear construction of integer tangles,the standard algebraic projection diagram of algebraic link and the corresponding standard weighted tree graph,the estimation of the stick index of a kind of algebraic links is obtained.