Topology Seminar

The Berge Conjecture asserts that any knot in $S^3$ admitting a lens space surgery has a surgery dual that is a simple knot in a lens space, and Berge provided twelve families of such knots. Greene and Berge independently proved that this list is complete among simple knots. In this dissertation, we consider an analogue of the Berge Conjecture: lens space surgeries arising from knots in other simple $3$-manifolds, specifically the Poincaré homology sphere and $S^1 \times S^2$. In the case of the Poincaré homology sphere, we develop a construction that produces infinitely many knots admitting lens space surgeries, recovering some results of Hedden and Tange. For $S^1 \times S^2$, we focus on the dual knots in the resulting lens spaces. Using the width of the Alexander polynomial, we constrain the possible homology classes of knots in lens spaces that admit $S^1 \times S^2$ surgeries. This approach plays an important role in completing the classification of BBL knots as the only simple knots in lens spaces admitting longitudinal $S^1 \times S^2$ surgeries. This is a Thesis Defense.