When will the crossing number of an alternating link decrease by two via a crossing change?

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Let D be a reduced alternating diagram of a non-split link L and L be the link whose diagram is obtained from D by a crossing change. If L is alternating, then c(L)≤ c(L)-2. In this paper we explore when c(L)=c(L)-2 holds and obtain a simple sufficient and necessary condition in terms of plane graphs corresponding to L. This result is obtained via analyzing the behavior of the Tutte polynomial of the signed plane graph corresponding to L.