The 6- and 8-palette numbers of links

For an effectively n-colorable link L, Cn⁎(L) stands for the minimum number of distinct colors used over all effective n-colorings of L. It is known that Cn⁎(L)≥1+log2⁡n for any effectively n-colorable link L with non-zero determinant. The aim of this paper is to prove that C6⁎(L)=4 and C8⁎(L)=5 for any effectively 6- and 8-colorable link L, respectively. For ribbon 2-links, we prove the same equalities for n=6 and 8, and C13⁎(L)=5 for n=13.