Heegaard Floer Surgery Formula and Cosmetic Surgeries
Abstract: Two Dehn surgeries on a knot are called cosmetic if they yield homeomorphic three-manifolds. We show for a certain family of null-homologous knots in any closed orientable three-manifold, if the knot admits cosmetic surgeries with a pair of positive surgery coefficients, then the coefficients are both greater than $1$. In addition, for this family of knots, we show that $1/q$ Dehn surgery for $q$ at least $2$ is not homeomorphic to the original three-manifold. The proofs of these results use the mapping cone formula for the Heegaard Floer homology of Dehn surgery in terms of the knot Floer homology of the knot; we provide a new proof of this formula for integer surgeries in $\text{Spin}c$ structures with nontorsion first Chern class.
- Jonathan Hanselman. Heegaard floer homology and cosmetic surgeries in S3superscript𝑆3S^{3}italic_S start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT. Journal of the European Mathematical Society, 2019.
- Rob Kirby. Problems in low-dimensional topology. 1995.
- Yi Ni. Thurston norm and cosmetic surgeries. arXiv: Geometric Topology, 2010.
- Yi Ni. Property g and the 4444–genus. arXiv: Geometric Topology, 2020.
- Holomorphic disks and knot invariants. Advances in Mathematics, 186(1):58–116, 2004.
- Holomorphic disks and topological invariants for closed three-manifolds. Annals of Mathematics, 159(3):1027–1158, 2004.
- Holomorphic triangles and invariants for smooth four-manifolds. Advances in Mathematics, 202(2):326–400, 2006.
- Knot Floer homology and integer surgeries. Algebraic & Geometric Topology, 8(1):101 – 153, 2008.
- Knot floer homology and rational surgeries. Algebraic & Geometric Topology, 11(1):1–68, January 2010.
- Ian Zemke. A general heegaard floer surgery formula, 2023.
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