ACHIRALITY AND LINKING NUMBERS OF LINKS

An oriented and ordered n-component link L in the 3-sphere is said to be achiral if it is ambient isotopic to its mirror image ignoring orientations and ordering of the components. For an oriented and ordered n-component link L, let λ_L be the product of linking numbers of all 2-component sublinks in L. For n = 4m + 3, where m is a non-negative integer, we show that if L is achiral then λ_L = 0. And for n ≠ 4m + 3, we show that there exists an n-component achiral link L with λ_L ≠ 0 by using achiral embeddings of complete graphs with n vertices K_n.