Given an n ‐component link L in any 3‐manifold M , the space L ⊂ ( Q ∪ { ∞ } ) n of rational surgery slopes yielding L‐spaces is already fully characterized in joint work by the author when n = 1 and L is nontrivial. For n > 1 , however, there are no previous results for L as a rational subspace, and only limited results for integer surgeries L ∩ Z n on S 3 . Herein, we provide the first nontrivial

Let K be a knot in a rational homology sphere M . This paper investigates the question of when modifying K by adding m > 0 half‐twists to two oppositely oriented strands, while keeping the rest of K fixed, produces a knot isotopic to K . Such a two‐strand twist of order m , as we define it, is a generalized crossing change when m is even and a non‐coherent band surgery when m = ± 1 . A cosmetic two‐strand

We produce for each tropical hypersurface V ( ϕ ) ⊂ Q = R n a Lagrangian L ( ϕ ) ⊂ ( C ∗ ) n whose moment map projection is a tropical amoeba of V ( ϕ ) . When these Lagrangians are admissible in the Fukaya–Seidel category, we show that they are unobstructed objects of the Fukaya category, and mirror to sheaves supported on complex hypersurfaces in a toric mirror.

We study a generalized cusp C that is diffeomorphic to [ 0 , ∞ ) times a closed Euclidean manifold. Geometrically, C is the quotient of a properly convex domain in R P n by a lattice, Γ , in one of a family of affine Lie groups G ( ψ ) , parameterized by a point ψ in the (dual closed) Weyl chamber for SL ( n + 1 , R ) , and Γ determines the cusp up to equivalence. These affine groups correspond to