Research On Coloring Of Several Classes Of 3-Tangles

Since the middle of the 20th century,the theory of three-dimensional manifolds in the direction of low-dimensional topology has developed rapidly and many important theoretical results have been obtained.The theory of knot and link is an important research branch of three-dimensional manifolds theory,it has always been an important subject in the study of knot and link theory that how to obtain the isotopy invariant of knot and link with strong distinguish ability and are easy to calculate through analysis and research.The invariants of knots and links also include Alexander polynomial,Conway polynomial,Jones polynomial,Crossing number,Arf invariant,Kauffman polynomial,Cross-cap number,Vassiliev polynomial,Homfly polynomial,and Linking number of knots and links,etc.Specifically,in 1928,American mathematician Alexander made a major breakthrough in the study of the invariant of knots and links,and gave Alexander polynomial invariant to links and konts;In 1969,Conway further studied Alexander polynomial invariants,then further analyzed and extended the related polynomials to obtain the improved Alexander polynomial;In 1970,J.H.Conway defined the continuous fraction of rational tangles,the classification of rational tangles is realized;In 2004,L.H.Kauffman and S.Lambropoulou defined the coloring rules of tangles,the classification of tangles is also realized by the method of color tangles.In this paper,it is based on tangle coloring and discusses the coloring and related properties of several classes of n-tangles(n? 3),combines the tangles with the basic polyhedron which corresponding to knots and links,then gives the definition of generalized tangles,and discusses the related properties of several classes of generalized tangles.In particularly,the specific coloring methods of several3-tangles are found,under these specific coloring methods,the coloring integers required for tangles are the least,and the coloring matrix has symmetrical,on this basis,it further discusses the coloring problem of n-tangles(n? 4).In addition,in this paper,the complete non-algebraic connected basic polyhedron with six vertexes are split,and the horizontal integer tangle is combined with the complete non-algebraic connected basic polyhedron to obtain several generalized tangles,then the coloring matrices of these kinds of generalized tangles are discussed and given by using the coloring rules of tangles,then on the basis of this,the coloring properties and the relationships between the coloring matrices are further given.Several kinds of tangles and their coloring properties in this paper provide a new research idea for the further study of the properties and classification of generalized tangles and more general n-tangles,it also gives a positive promotion for the classification of algebraic knots and links.