Charles Delman (EIU Math) Alternating knots and Montesinos knots satisfy the Lspace knot conjecture. Joint work with Rachel Roberts Abstract: An Lspace is a homology \(3\)sphere whose HeegardFloer homology has minimal rank; lens spaces are examples (hence the name). Results of Ozsváth  Szabó, Eliashberg Thurston, and Kazez  Roberts show that a manifold admitting a taut, coorientable foliation cannot be an Lspace. Let us call such a manifold foliar. Ozsváth and Szabó have asked whether or not the converse is true for irreducible \(3\)manifolds; Juhasz has conjectured that it is. Restricting attention to manifolds obtained by Dehn surgery on knots in \(S^3\), we posit the following: Lspace Knot Conjecture. Suppose \( \kappa \subset S^3\) is a knot in the 3sphere. Then a manifold obtained by Dehn filling along \(\kappa\) is foliar if and only if it is irreducible and not an Lspace. Using generalized surface decomposition techniques that build on earlier work of Gabai, Menasco, Oertel, and the authors, we prove that both alternating knots and Montesinos Knots satisfy the Lspace Knot Conjecture. We believe these techniques will prove fruitful in the further study of taut foliations in \(3\)manifolds. 
