Abstract
The classification of knots in closed 3–manifolds is a central theme in low–dimensional topology. While knots in (S^3) are strongly constrained by the Gordon–Luecke knot complement theorem and Thurston–Perelman geometrization, the structure of hyperbolic knots in lens spaces remains comparatively less understood. Knots in lens spaces arise naturally as surgery duals of knots in (S^3) admitting lens space surgeries, placing them within the broader Dehn surgery classification problem.This dissertation investigates constraints for hyperbolic knots in lens spaces through the topology of rational Seifert surfaces. If (K\subset L(p,q)) represents a torsion class of order (s), a rational Seifert surface is a properly embedded oriented surface (S\subset E(K)) whose boundary represents (s) times a longitudinal class. Hyperbolicity excludes essential disks and annuli, but punctured spheres of Euler characteristic (-1) and (-2) are not ruled out by general geometric principles alone.The main results are two theorems establishing that a non–null–homologous hyperbolic knot in a lens space admits no rational Seifert surface homeomorphic to a three–punctured sphere for all lens spaces or a four–punctured sphere for those lens spaces admitting no embedded Klein bottles. The proofs adapt intersection graph techniques and Scharlemann cycle analysis. Intersections of (S) with a punctured Heegaard torus (T) are encoded by labeled graphs (G_S\subset S) and (G_T\subset T). Disk faces in (G_S) force Scharlemann cycles, yielding controlled solid–torus and annulus structures. Interaction arguments then force either essential annuli or tori in the knot exterior, contradicting hyperbolicity, or embedded nonorientable surfaces in the ambient lens space, contradicting the Klein-bottle exclusion in the four–punctured sphere case.These results extend prior work on knots in lens spaces admitting low–complexity surfaces and support the broader picture that hyperbolic knots exhibit stronger rigiditywith their existence and structure tightly constrained by the rational Seifert surfaces.