Knot concordance in homology cobordisms

A surgery formula for knot Floer homology

“…knots in S 3 ), any map realizing the i $ j symmetry suffices. (This principle has been used, implicitly or explicitly, by many authors; see, e.g., [16,Section 6]. )…”

“…This formula has been one of the most important tools in the Heegaard Floer toolkit. Not only has it has been the primary method of computation for many specific examples of Floer homology groups [2,8,12,16,19,24,30], but the existence of the formula indicates that the knot Floer homology invariants tightly constrain the Floer invariants of manifolds obtained by surgery, and conversely. This interplay between the two invariants, coupled with the rich geometric content of both, has led to striking new applications in Dehn surgery.…”

“…Most recently, Hom, Lidman, and Levine [16] have used our main theorem (Theorem 1.1) to provide an example of a knot in a homology sphere which has infinite order in the non-locally-flat piecewise-linear concordance group. The reader is encouraged to refer to that paper for a detailed computation using this formula, which illustrates the general technique.…”

Contact $(+1)$-surgeries on rational homology $3$-spheres

“…Then we introduce the definition of rational ν invariants in the same manner as Hom-Levine-Lidman did in the integral homology sphere case ( [24]).…”

PL-Genus of surfaces in homology balls

“…Let be a genus g PL surface in an integer homology ball X , such that . Up to isotopy, we may assume that is smooth, except at finitely many singular points, each of which is modeled on the cone of a smooth knot in (see, for example, [4, Theorem A.1]). By deleting neighborhoods of arcs in connecting the cone points, we obtain a genus g cobordism from the knot to K in a homology cobordism from to Y .…”