Effect of a crossing change on crossing number
  The purpose of this article is to give a preliminary clarification on the relation between crossing number and crossing change. With a main focus on the span of X polynomial, we prove that, as our theorem claims, the crossing number of the link after crossing change can be estimated when certain conditions are met. At the end of the article, we give an example to demonstrate a special case for the theorem and a counterexample to explain that the theorem cannot be applied if the obtained link is not alternating. 
Longting Wu, Shuting Shao, Shan Liu, Fengchun Lei
math.GT