| Knot theory is an important branch of low-dimensional topology.It mainly studies closed non-self-crossing curves in three-dimensional Euclidean space.The basic problem of knot theory is the classification of knots and links.In 1970,through the research of tangles,J.H.Conway gave a new important method for studying the knot theory,and used the correspondence between rational tangle and consecutive fractions to prove the equivalence of two rational tangles.The more in-depth research and discover hidden algebraic structures of tangles,the better processing of tangle classification can be,and thus the more helpful for the classification of knots and links.In the past,experts and scholars in the low-dimensional topological direction at home and abroad have made additions and multiplications from rational tangles,and then proceeded to construct algebraic knots and links,and systematically studied and analyzed the nature of the resulting knots or links.By knots and links invariants,such as the linking number,crossing number,component number and sticks index,the experts and scholars gave many meaningful results.In this master thesis,starting from the rational tangle of the decomposition of algebraic knots and links,it use the stick index invariants of tangle to further explore the properties of algebraic knots and links.In the master thesis,a kind of rational tangle is multiplied or added to get algebraic tangle,then a polygonal representation of the algebraic tangle is constructed,and finally polygonal representation of the algebraic knots and links are obtained.Furthermore,by the same method,it give an estimate of the number of sticks in an algebraic knot or link under normal conditions.The method of this master thesis proves a broad idea for research and lays a further research foundation for algebraic knots and links.In addition,this master thesis further discusses and gives the tangle sticks index relationship between non-primed tangle and its factors. |
PDF Full Text Request