The Polygon Representation Of Pretzel Link

It is well known that isotopy classification of knots and links is a central topic in knottheory, and the various invariants of knots and links are important tools for the isotopyclassification of knots and links. Commonly used invariants are Jones polynomial, Alexanderpolynomial, bridge number, superbridge number and so on.Based on knot theory research, it is not difficult to find that the stick number of knotsand links is also an important invariant in the isotopy classification of knots and links. Thereare some important connections between this invariant and other isotopy invariants. Theseconnections play an important role in the isotopy classification in which various types ofinvariant of knot and link are used correspondingly. Pretzel link is a very important class inlinks. Therefore, it is a very meaningful work to use stick number in the isotopy classificationof Pretzel links.In this paper, we give a stick representation of Pretzel knots by the stick representationinteger tangles, and get a estimation of the stick number by crossing number of Pretzel knots.By discussing and analyzing the component number of Pretzel links, the conditions that aPretzel link is a knot is determined, and if a pretzel link is a knot, there are only two class ofthe knots. I.e, Pretzel knot and the generalized Pretzel knot. Using the same method, the sticknumber of the generalized Pretzel knots are also estimated. We get the stick number of somespecific simple pretzel links by applying the estimation about the stick number of Pretzel links.It is indicated that the estimation of stick number of Pretzel links in this paper is moreexcellent in determining the stick number for certain knots. Two components Pretzel link is aparticular class in Pretzel links. There are only two types of two components Pretzel link bydiscussing the component number of Pretzel links. Using the stick representation of tangles,we estimate the stick number of two components Pretzel link.