Cosmetic surgeries on knots in $S^3$

@article{Ni2010CosmeticSO,
  title={Cosmetic surgeries on knots in \$S^3\$},
  author={Yi Ni and Zhongtao Wu},
  journal={arXiv: Geometric Topology},
  year={2010}
}
  • Yi Ni, Zhongtao Wu
  • Published 23 September 2010
  • Mathematics, Medicine
  • arXiv: Geometric Topology
Two Dehn surgeries on a knot are called {\it purely cosmetic}, if they yield manifolds that are homeomorphic as oriented manifolds. Suppose there exist purely cosmetic surgeries on a knot in $S^3$, we show that the two surgery slopes must be the opposite of each other. One ingredient of our proof is a Dehn surgery formula for correction terms in Heegaard Floer homology. 
Connected sums of knots do not admit purely cosmetic surgeries
Two Dehn surgeries on a knot are called purely cosmetic if their surgered manifolds are homeomorphic as oriented manifolds. Gordon conjectured that non-trivial knots in $S^3$ do not admit purely
3-braid knots do not admit purely cosmetic surgeries
A pair of surgeries on a knot is called purely cosmetic if the pair of resulting 3-manifolds are homeomorphic as oriented manifolds. An outstanding conjecture predicts that no nontrivial knots admit
Cable knots do not admit cosmetic surgeries
  • R. Tao
  • Mathematics
    Journal of Knot Theory and Its Ramifications
  • 2019
Two Dehn surgeries on a knot are called purely cosmetic if their surgered manifolds are homeomorphic as oriented manifolds. Gordon conjectured that nontrivial knots in [Formula: see text] do not
On The Virtual Cosmetic Surgery Conjecture
Let K be a knot in S^3, and M and M' be distinct Dehn surgeries along K. We investigate when M covers M'. When K is a torus knot, we provide a complete classification of such covers. When K is a
A note on Jones polynomial and cosmetic surgery
We show that two Dehn surgeries on a knot $K$ never yield manifolds that are homeomorphic as oriented manifolds if $V_K''(1)\neq 0$ or $V_K'''(1)\neq 0$. As an application, we verify the cosmetic
Chirally Cosmetic Surgeries and Casson Invariants
We study chirally cosmetic surgeries, that is, a pair of Dehn surgeries on a knot producing homeomorphic 3-manifolds with opposite orientations. Several constraints on knots and surgery slopes to
Heegaard Floer homology and chirally cosmetic surgeries
TLDR
This work completely classify chirallly cosmetic surgeries on odd alternating pretzel knots, and rule out such surgeries for a large class of Whitehead doubles.
Dehn Surgery on Knots in S3 Producing Nil Seifert Fibered Spaces
We prove that there are exactly 6 Nil Seifert fibred spaces which can be obtained by Dehn surgeries on non-trefoil knots in S^3, with {60, 144, 156, 288, 300} as the exact set of all such surgery
On constraints for knots to admit chirally cosmetic surgeries and their calculations
We discuss various constraints for knots in S3 to admit chirally cosmetic surgeries, derived from invariants of 3-manifolds, such as, the quantum SO(3)-invariant, the rank of the Heegaard Floer
Purely cosmetic surgeries and pretzel knots
We show that all pretzel knots satisfy the (purely) cosmetic surgery conjecture, i.e. Dehn surgeries with different slopes along a pretzel knot provide different oriented three-manifolds.
...
...

References

SHOWING 1-10 OF 22 REFERENCES
Cosmetic surgeries on genus one knots
In this paper, we prove that there are no truly cosmetic surgeries on genus one classical knots. If the two surgery slopes have the same sign, we give the only possibilities of reflectively cosmetic
Cosmetic Surgery in Integral Homology $L$-Spaces
Let $K$ be a non-trivial knot in $S^3$, and let $r$ and $r'$ be two distinct rational numbers of same sign, allowing $r$ to be infinite; we prove that there is no orientation-preserving homeomorphism
CLOSED 3–MANIFOLDS UNCHANGED BY DEHN SURGERY
If X is the space of the trefoil knot, we can find, for every integer e, two slopes on ∂X, ρe, γe, such that the Dehn fillings X (ρe) and X (γe) are homeomorphic 3-manifolds. The cores of the
Dehn surgery on knots
Knot Floer homology and integer surgeries
Let Y be a closed three-manifold with trivial first homology, and let K Y be a knot. We give a description of the Heegaard Floer homology of integer surgeries on Y along K in terms of the filtered
Knot Floer homology and rational surgeries
Let K be a rationally null-homologous knot in a three-manifold Y . We construct a version of knot Floer homology in this context, including a description of the Floer homology of a three-manifold
Holomorphic disks and knot invariants
On the Floer homology of plumbed three-manifolds
We calculate the Heegaard Floer homologies for three-manifolds obtained by plumbings of spheres specified by certain graphs. Our class of graphs is sufficiently large to describe, for example, all
Knot Floer homology and the four-ball genus
We use the knot filtration on the Heegaard Floer complex to define an integer invariant tau(K) for knots. Like the classical signature, this invariant gives a homomorphism from the knot concordance
Floer homology and knot complements
We use the Ozsvath-Szabo theory of Floer homology to define an invariant of knot complements in three-manifolds. This invariant takes the form of a filtered chain complex, which we call CF_r. It
...
...