Finiteness Of Involutory Quandles Of Some Knots

The application of quandle theory in knot theory contacts aspects of topology,algebra and combinatorics.Knot quandles are the complete invariants of unoriented knots,but usually it is more convenient to deal with the problems by con-sidering the specific quotient,i.e.involutory quandles.We can obtain the presentation of involutory quandle from each knot projection,and then draw the Cayley graph including some algebraic information about the involutory quandle.In addition,the problem of the finiteness of the knot involutory quandles is still being explored.The purpose of this paper is to study a problem,the finiteness of the involutory quandle of rational knots and pretzel knots.In order to prove that the involutory quandles of rational knots are finite,in addition to enumerating the elements of the involutory quandles directly by using the Schubert standard forms of the rational knots,the other method is to use the lemma that if the involutory quandle of knot is abelian,then it is finite,what we have done is to prove that the involutory quandles of rational knots are abelian,and we get the same conclusion.Although for the projection of each pretzel knot we can draw its Cayley graph of involutory quandle,it is not convenient for us to judge whether the involutory quandle is finite,so we use the concept of circle rank in graph theory to prove the conclusion:The involutory quandle of the pretzel knot P(p1,p2,p3)(Suppose pi,p2,P3 are all integers greater than 1)is finite if and only if 1/p1+1/p2+1/p3>1.