Representations Of Generalized Weighted Tree Graphs Of Several Kinds Of Non-MAP Links And Estimation Of Stick Index Of A Class Of Algebraic Links

low-dimensional topology is an important branch of topology,the theory of knots and links is an important part of low-dimensional topology,the classification of knots and links is a core content of the combination theory of three-dimensional manifolds.Many important properties of knots and links can be obtained through elementary transformations and various constructions,at the same time,the relationships between knots and links can be found,which can be used to achieve the classification of knots and links and simplify the research process.Up to now,mathematicians at home and abroad have classified knots and links from many different perspectives,and they gave the estimation of the stick index Ls)(of many kinds of knots and links.In 1970,from the perspective of decomposition of tangles,J.H.Conway pointed out in reference[1] that any projection diagrams of knots and links which can be obtained by algebraic tangles makes the connection.On this basis,he defined the algebraic links and introduced the relationships between the projection diagrams of knots and links and their basic polyhedrons;In 2004,in the reference[2],C.L.Mc Cabe gave the concrete method of transforming the projection diagrams of algebraic links into standard form,he defined MAP links and the standard weighted tree graph representations of MAP links,he defined the piecewise linear construction models of integer tangles and established the one-to-one correspondence between knots and links and their tree graphs.On this basis,he got the estimation of stick number of fans;In 1991,in the reference [2],S.Negami established the relationships between the stick index of knots and links and crossing number,and gave an estimation of the stick index of any knots and links expect for the hopf link.In this paper,from the perspective of projection diagrams of knots and links,the influence of R transformations on the weighted tree graphs of projection diagrams of knots and links are proved,and the relationships between the tree graphs before and after transformations are obtained by using the research techniques and methods of the combinatorial topology of three-dimensional manifolds;And it is proved that the self-connected sum constructions of knots and links have effects on the projection diagrams and their corresponding tree graphs;On the basis of the standard weighted tree graphs of MAP links,this paper constructs the representations of generalized weighted tree graphs of several kinds of non-MAP links.According to the piecewise linear construction models of integer tangles,it obtains the estimation of stick index of a class of algebraic links.The results are obtained in this paper have some theoretical significance for the further classification and property research of knots and links.