# Small Dehn surgery and SU(2)

@inproceedings{Baldwin2021SmallDS, title={Small Dehn surgery and SU(2)}, author={John A. Baldwin and Zhenkun Li and Steven Sivek and Fan Ye}, year={2021} }

We prove that the fundamental group of 3-surgery on a nontrivial knot in S always admits an irreducible SU(2)-representation. This answers a question of Kronheimer and Mrowka dating from their work on the Property P conjecture. An important ingredient in our proof is a relationship between instanton Floer homology and the symplectic Floer homology of genus-2 surface diffeomorphisms, due to Ivan Smith. We use similar arguments at the end to extend our main result to infinitely many surgery…

## 4 Citations

### Khovanov homology and the cinquefoil

- Mathematics
- 2021

We prove that Khovanov homology with coefficients in Z/2Z detects the (2, 5) torus knot. Our proof makes use of a wide range of deep tools in Floer homology, Khovanov homology, and Khovanov homotopy.…

### Knot surgery formulae for instanton Floer homology II: applications

- Mathematics
- 2022

This is a companion paper to earlier work of the authors, which proved an integral surgery formula for framed instanton homology. First, we present an enhancement of the large surgery formula, a…

### An instanton take on some knot detection results

- Mathematics
- 2022

. We give new proofs that Khovanov homology detects the ﬁgure eight knot and the cinquefoils, and that HOMFLY homology detects 5 2 and each of the P ( − 3 , 3 , 2 n + 1) pretzel knots. For all but…

### Instanton Floer homology, sutures, and Euler characteristics

- Mathematics
- 2021

This is a companion paper to an earlier work of the authors. In this paper, we provide an axiomatic definition of Floer homology for balanced sutured manifolds and prove that the graded Euler…

## References

SHOWING 1-10 OF 44 REFERENCES

### Toroidal homology spheres and SU(2)-representations

- Mathematics
- 2021

We prove that if an integer homology three-sphere contains an embedded incompressible torus, then its fundamental group admits irreducible SU(2)representations. Our methods use instanton Floer…

### Dehn surgery, the fundamental group and SU(2)

- Mathematics
- 2003

Let K be a non-trivial knot in the 3-sphere and let Y(r) be the 3-manifold obtained by surgery on K with surgery-coefficient a rational number r. We show that there is a homomorphism from the…

### Stein fillings and SU(2) representations

- MathematicsGeometry & Topology
- 2018

We recently defined invariants of contact 3-manifolds using a version of instanton Floer homology for sutured manifolds. In this paper, we prove that if several contact structures on a 3-manifold are…

### Witten's conjecture and Property P

- Mathematics
- 2003

Let K be a non-trivial knot in the 3{sphere and let Y be the 3{manifold obtained by surgery on K with surgery-coecient 1. Using tools from gauge theory and symplectic topology, it is shown that the…

### Khovanov homology and the cinquefoil

- Mathematics
- 2021

We prove that Khovanov homology with coefficients in Z/2Z detects the (2, 5) torus knot. Our proof makes use of a wide range of deep tools in Floer homology, Khovanov homology, and Khovanov homotopy.…

### Symmetry of knots and cyclic surgery

- Mathematics
- 1992

If a nontorus knot K admits a symmetry which is not a strong inversion, then there exists no nontrivial cyclic surgery on K. No surgery on a symmetric knot can produce a fake lens space or a…

### Instanton L-spaces and splicing

- Mathematics
- 2021

. We prove that the 3-manifold obtained by gluing the complements of two nontrivial knots in homology 3-sphere instanton L -spaces, by a map which identiﬁes meridians with Seifert longitudes, cannot…

### Instanton Floer homology and the Alexander polynomial

- Mathematics
- 2010

The instanton Floer homology of a knot in the three-sphere is a vector space with a canonical mod 2 grading. It carries a distinguished endomorphism of even degree,arising from the 2-dimensional…

### A Menagerie of SU(2)-Cyclic 3-Manifolds

- MathematicsInternational Mathematics Research Notices
- 2021

We classify $SU(2)$-cyclic and $SU(2)$-abelian 3-manifolds, for which every representation of the fundamental group into $SU(2)$ has cyclic or abelian image, respectively, among geometric…

### On families of fibred knots with equal Seifert forms

- MathematicsCommunications in Analysis and Geometry
- 2021

For every genus $g\geq 2$, we construct an infinite family of strongly quasipositive fibred knots having the same Seifert form as the torus knot $T(2,2g+1)$. In particular, their signatures and…