The Constructions And Polygonal Representations Of Several Classes Of Knots And Links

The core problem of knot and link theory is to find invariants that can classify the knots and links well and have easy calculation,the stick number of knots and links is one of them.So far,mathematicians at home and abroad have determined and estimated the stick number of various knots and links from variety of different perspectives.In 1991,S.Negami established the stick number of knots and links with the crossing number,and gave an estimate of the stick number of any knots and links other than the Hopf link is(?).In 2002,J.Calvo further refined the lower bound of this inequality and proved that s(K)is less than or equal to(?).in 2011,Y.Huh and S.Oh gave a more accurate upper bound from the perspective of combinatorial construction,and proved that the stick number of any non-trivial knots K is less than or equal to 3/2(C(K)+1),and when K is a non-trivial alternating knot,the upper bound on stick number of K is 3/2c(K).In this paper,from the perspective of knot and link projections,first of all,it constructively proves the inclusive existence of a class of knots and links which are consisted of n(n?6,n?7,n ?N+)algebraic tangles in a completely non-algebraic connection by using the research techniques and methods of three-dimensional manifold and combinatorial topology.Further,the upper bounds on the stick number for several classes of algebraic and non-algebraic knots and links are given in this paper.And it also proves when the crossing number c(K)satisfies some certain conditions,the upper bounds on such non-algebraic knots and links are slightly more accurate than the results given in 2005 by E.Insko.In summary,the research results of this paper have certain guiding significance for further research on the stick number of non-algebraic knots and links and the coloring properties of non-algebraic tangles.