Minimum numbers of Dehn colors of knots and symmetric local biquandle cocycle invariants

This is the second paper which discusses minimum numbers of “region” colors for knots, while minimum numbers of arc colors are well-studied, where the first one is our previous paper [E. Matsudo, K. Oshiro and G. Yamagishi, Minimum numbers of Dehn colors of knots and [Formula: see text]-palette graphs, Adv. Appl. Math. 173 (2026) 102995]. In this paper, we give a method to evaluate minimum numbers of Dehn colors for knots by using symmetric local biquandle cocycle invariants. We give answers to some questions arising as a consequence of our previous paper [E. Matsudo, K. Oshiro and G. Yamagishi, Minimum numbers of Dehn colors of knots and [Formula: see text]-palette graphs, Adv. Appl. Math. 173 (2026) 102995]. In particular, we show that there exist knots which are distinguished by minimum numbers of Dehn colors.