Analogues in other 3-manifolds for thinness and SL(2,C)-abelian surgeries

Determine whether analogues of Theorem 1.1 (characterization of knots with infinitely many SL(2,C)-abelian surgeries) and Theorem 1.2 (thin knots are exactly torus knots) hold for knots in closed 3-manifolds other than S^3; in particular, establish whether a knot K⊂Y with irreducible complement is thin, or admits infinitely many SL(2,C)-abelian surgeries, if and only if its complement is Seifert fibered over the disk with at most two singular fibers.

Background

The authors extend their viewpoint beyond S^3, asking whether their main theorems on thinness and SL(2,C)-abelian surgeries admit counterparts for knots in general 3-manifolds. They highlight Seifert fibered complements over the disk with few singular fibers as the expected characterization in this broader setting.

Such results would generalize the classification of thin knots and the behavior of representation varieties of knot complements, potentially linking A-polynomial features and instanton Floer constraints in more general ambient manifolds.