On the relation between Betti numbers of an Arf semigroup and its blowup

In this note we prove the relation between Betti numbers of an Arf semigroup $S$ and its blowup $S^{\mathrm{\prime }}$ in the case when they have the same multiplicity $n$. The relation is then ${\beta }_{i,s}(S^{\mathrm{\prime }})={\beta }_{i,s+(i+1)n}(S)$.