Surface-links represented by 4-charts and quandle cocycle invariants, II

In this paper, we give lower bounds of the braid indices of surface-links by represented 4-charts by quandle cocycle invariants. As a consequence, we show that the braid index of the connected sum of a spun trefoil and a 2-twist spun trefoil is five, which is a special case of Tanakaʼs Problem.