For Artin groups of dihedral type, we compute the Bredon homology classifying space for family virtually cyclic subgroups with coefficients in K-theory a group ring.

Let $D_4$ be the dihedral group of order $8$. In present study, we give generators algebra invariants in polynomial with four over a field characteristic zero.

Under the assumption that a defining graph of Coxeter group admits only subsequent elementary twists in $$\mathbb {Z}_2$$ or dihedral groups and is type $$\mathrm {FC}$$ , we prove Bernhard Mühlherr’s Twist Conjecture.