The Coloring Properties Of Several Classes Of Symmetrical 2-Tangles And A Class Of N-Tangles

A core content of knot and link theory research is to find some invariants that can distinguish knots and links well and are easy to calculate.These invariants include Jones polynomial,Alexander polynomial,stick index and so on.In 1970,J.H.Conway defined the continued fraction of rational tangles and used it as a tool to achieve the classification of rational tangles.In 2004,L.H.Kauffman and S.Lambropoulou defined the coloring rule of tangles and found that the coloring fraction is also an important invariant to distinguish rational tangles.As we all know,knots and links can be obtained from the D construction or N construction of tangles,so studying the coloring properties of tangles is of great significance to the classification of knots and links.From the perspective of the coloring of rational tangles,this paper gives the general form and related conclusions of the color matrices of rational tangles,and further explores the coloring properties of several classes of symmetrical 2-tangles and a class of n-tangles by using the research methods and techniques of three-dimensional manifold and combinatorial topology.Specifically,this paper explores the coloring properties of several classes of horizontally symmetrical tangles,vertically symmetrical tangles and diagonally symmetrical tangles,as well as some of their generalizations,and through further analysis,calculation,it discovers the regularity of the color matrices of several special classes of symmetrical 2-tangles in general.In addition,combined with the coloring of 2-tangles,this paper gives the color matrix of a special class of reticulated n-tangles under certain limited conditions,and attempts to convert the questions on the coloring and classification of complex tangles into the questions on the coloring and classification of their corresponding sub-tangles.Finally,this paper extends the diagonal sum rule of color matrices of 2-tangles to the color matrices of a class of n-tangles,and obtains the diagonal sum rule of color matrices of this class of n-tangles.The coloring properties of several classes of tangles obtained in this paper not only provide a new idea and approach for further research on the coloring properties and classification of more general algebraic tangles and n-tangles,but also provide a certain direction for subsequent research on the coloring properties of general tangles.