Estimations Of Stick Index Of Several Kinds Of Algebraic Links And Generalized Weighted Tree Graphs Of A Class Of Spatial Graphs

Low-dimensional topology is an important branch of topology,and the theory of knot and link is a significant research content of low-dimensional topology.The isotopy classification of knots and links is a core problem in the research of knot and link theory,and various types of isotopy invariants of knots and links are important tools for the isotopy classification of knots and links.The invariants of knots and links such as Jones polynomial,Alexander polynomial,Kauffman polynomial,bridge index,superbridge index,crossing index,stick index and so on are usually used to classify knots and links.Among them,the stick index of knots and links is an important invariant to study properties and classifications of knots and links from the perspective of combinatorial topology of three-dimensional manifold and using the research techniques and methods of combinatorial topology of three-dimensional manifold.So far,experts and scholars at home and abroad have given a lot of stick index estimations of knots and links from a variety of different perspectives.In this thesis,by using the techniques and methods of combinatorial topology theory of three-dimensional manifold,the stick index estimations of several kinds of algebraic links are studied.Firstly,the possible positions of stumps with weight of ?1 are analyzed and discussed in the corresponding weighted tree graphs of some algebraic links projections.The stick index estimation of the corresponding algebraic links is given by adjusting the angles of the exit edges for the existing polygonal representations of local tangles and reducing the number of edges to construct tangles polygonal representations.Secondly,for the positions of stumps with weight of ?2 and the possible situations of weight of adjacent stumps in the corresponding weighted tree graphs of some algebraic links projections are analyzed and discussed,the constructions of polygonal representations of-? 2 tangles in different positions are adjusted,and the angles of the exit edges of the existing polygonal representations for other related tangles are further adjusted,thus the stick index estimations of several kinds of algebraic links are given.On this basis,this thesis analyzes and discusses the case that the weights of stumps in the corresponding weighted tree graphs of the algebraic links are all ?2 and the tangles of the corresponding subtree graphs are all rational tangles,the polygonal representations of-? 2 tangles are adjusted appropriately according to the corresponding rules,and the stick index estimation of this kind of algebraic links is obtained.Finally,generalized weighted tree graphs of a class of approximable algebraic representation of spatial graphs are given.The research methods and results of stick index estimations of some algebraic links in this thesis provide a certain direction for further research of stick index estimations of general algebraic links and spatial graphs.