Polygonal Representations And Stick Index Of A Class Of Links

Knot and link invariants are always the core and main content of knot and link theory.Through the research of knot and link invariants,research scholars can give some important properties and classifications of knots and links.Stick index is a knot and link invariant.In recent years,the knot and link research scholars at home and abroad give many stick index results.Many results are as follows:In 1991,S.Negami presents a nontrivial knot stick index relationship with the crossing number;In 1993,G.T.Jin proves that the stick index of every nontrivial knot must be greater than or equal to 6;In 1994,R.Randell proves that the stick index of trefoil knot is equal to 6,the stick index of figure-eight knot is equal to 7.The sum of the stick index of knotK₁ and the stick index of knotK₂must be greater than or equal to the stick index of the composition of two knots;In 1997,C.Adams gives relationship between stick index of non-prime knot and other index;In 1997,G.T.Jin proves stick index of torus knot;In 2011,Y.Huh and S.Oh use the relationship between arc index and crossing number to further give the relationship between stick index of nontrivial knot and crossing number of nontrivial knot.Knots and links can be researched by tangle decompositions.This master's thesis estimates the stick index of a class of knots and links by the tangle decompositions of knots and links.In fact,rational tangle is made up of integer tangles which are obtained by addition and multiplication,so rational tangle stick index can be estimated on the basis of integer tangle stick index by combining two tangle central axis into a new central axis.This master's thesis gives the rational tangle polygon representations under different conditions,therefore the estimation of knots and links stick index can be given through N structure.The proof methods and results of this master's thesis will be a positive guidance for further discussion and estimation of the stick index of knots and links.