Tree Diagram Representation Of Several Classes Of Links And Their Related Properties

Link theory has a long history of development.It is developed as an important branch of topology in the 20 th century.In general,the common classification of knots is the basic problem of link theory.From the perspective of strict terminology,topology is the subject of researching continuous deformation of geometric figure,while link theory is the topology branch of researching the invariants of link under continuous deformations.Therefore,finding the invariants of link is the core problem of link theory.It is well known that stick number is one of the important invariants of link.In 1970,Conway points out that the projection of any link can be regarded as a collection of algebraic tangles,gives the definition of algebraic link and introduces the relationship between link projection and its basic polyhedron,which provides an effective method for further researching on stick number of link;in 1991,S.Negami gives the relationship between stick number and crossing number of non-trivial link other than the Hopf Link;in 1993,G.T.Jin gives the relationship between stick number and crossing number of torus knot;in 1998,C.L.McCabe gives the relationship between stick number and crossing number of two-bridge link;in 2011,Y.Huh and S.Oh give a better relationship between stick number and crossing number of algebraic link which forms by intergral tangles with crossing number greater than or equal to seven;in addition,D.Gabai gives the definition of arborescent link(i.e.,algebraic link)and the specific method of representing an oriented arborescent link with a special tree diagram;C.L.McCabe gives the method of transforming the projection of algebraic link into standard form on the basis of specific expression of algebraic link,defines piecewise-linear construction of algebraic link,and gives an estimation of stick number of algebraic link.From the perspective of studying link projection,combining the basic concepts of graph theory and the properties of tangle decomposition,this master thesis mainly studies tree diagram representation of several classes of links,and further exploits stick number estimation of special non-algebraic link.Specifically,this master thesis firstly uses the construction method of edge-connected 4-valent planar map to give the edge-connected4-valent planar map representation of a special class of non-algebraic link projection.Secondly,it gives the standard weighted tree diagram representation of algebraic link,and a spacific method of turning any weighted tree diagram into standard weighted tree diagram.Finally combining the edge-connected 4-valent planar map representation of link projectionand the definition and construction method of standard weighted tree diagram,the definition and construction method of generalized standard weighted tree diagram of non-algebraic link are given,meanwhile,the specifiied form of generalized standard weighted tree diagram of non-algebraic link which composed of six intergral tangles and its stick number estimation are also given.