PSL2(C)-character varieties and Seifert fibered cosmetic surgeries

We study small Seifert possibly chiral cosmetic surgeries on not necessarily null-homologous knot in rational homology spheres. We define an absolute semi-norm derived from the Culler-Shalen semi-norm of PSL2(C)-character variety theory. We prove that two cosmetic slopes must have the same absolute semi-norm and use this result to give a sharp bound on the number of slopes producing the same small Seifert manifold if the ambient manifold satisfies some representation theoretic conditions.