On the Ozsváth-Szabó $d$-invariants for almost simple linear graphs

Abstract: Karakurt and \c{S}avk calculate the Ozsv\'ath-Szab\'o $d$-invariant for Brieskorn homology $3$-spheres where surgery diagrams can be expressed as almost simple linear graphs. In this paper, we introduce a refiniment of Karakurt and \c{S}avk's formula for these $d$-invariants. We also introduce infinite examples of inequalities that appear in this refinement in which the equality holds, and infinite examples in which it does not hold. We also discuss an application to knot concordance group.