Unknotting Numbers Of Certain Types Of Pretzel Knots

The essential problem of the knot theory is how to distinguish distinct knots or links,while the knot invariants are major tools for judging whether or not two knots or links are equivalent.There are many knot invariants:such as crossing number,bridge number,unknotting number,braid index,genus and knot polynomials.Among them,unknotting number of non-trivial knot K is the minimal number of crossing changes required to transform K into the unknot.The methods used to determine the unknotting number of a knot are mainly the Heegaard Floer homology,the Alexander Polynomial Surgery,the Jones Polynomial,the R-moves and the isotopy.In this paper,the unknotting numbers of certain types of pretzel knots are studied.Based on the embedding representations of links,by using pass replace-ments,we obtain the unknotting numbers of certain new types of pretzel knots,which give the upper bounds of these pretzel knots in R3 and provide a new idea to prove the unknotting number.The main results are as follows:The paper gives the unknotting numbers of the univariate pretzel knot P(3,3,c)and P(3,5,c).Based on this,we give the unknotting number of new types bivari-ate pretzel knot P(a2-2,1,a3-2),and use this conclusion to study the unknotting number of P(1,a2,a3).Based on this conclusion,we get the unknotting number of P(a1,a2,a3)when a3 ≥ a2 ≥ a1 ≥ 1 and ai(1 ≤ i ≤ 3)is odd.Under the condition of deleting a3>a2>a1,the conclusion will be further extended.In addition,the unknotting number of P(3,b,2)are given for each positive integer b.