If a knot K in S^3 admits a pair of truly cosmetic surgeries, we show that the surgery slopes are either ±2 or ±1/q for some value of q that is explicitly determined by the knot Floer homology of K. Moreover, in the former case the genus of K must be 2, and in the latter case there is a bound relating q to the genus and the Heegaard Floer thickness of K. As a consequence, we show that the cosmetic crossing conjecture holds for alternating knots (or more generally, Heegaard Floer thin knots) with genus not equal to 2. We also show that the conjecture holds for any knot K for which each prime summand of K has at most 16 crossings; our techniques rule out cosmetic surgeries in this setting except for slopes ±1 and ±2 on a small number of knots, and these remaining examples can be checked by comparing hyperbolic invariants. These results make use of the surgery formula for Heegaard Floer homology, which has already proved to be a powerful tool for obstructing cosmetic surgeries; we get stronger obstructions than previously known by considering the full graded theory. We make use of a new graphical interpretation of knot Floer homology and the surgery formula in terms of immersed curves, which makes the grading information we need easier to access.
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Cusp volumes of alternating knots on surfaces
We study the geometry of hyperbolic knots that admit alternating projections on
embedded surfaces in closed 3–manifolds. We show that, under mild hypothesis, their
cusp area admits two-sided bounds in terms of the twist number of the alternating
projection and the genus of the projection surface. As a result, we derive diagrammatic
estimates of slope lengths and give applications to Dehn surgery. These generalize
results of Lackenby and Purcell about alternating knots in the 3–sphere.
Using a result of Kalfagianni and Purcell, we point out that alternating knots on
surfaces of higher genus can have arbitrarily small cusp density, in contrast to
alternating knots on spheres whose cusp densities are bounded away from zero
due to Lackenby and Purcell.
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- Award ID(s):
- 2004155
- NSF-PAR ID:
- 10417826
- Date Published:
- Journal Name:
- Algebraic geometric topology
- Volume:
- 22
- ISSN:
- 1472-2747
- Page Range / eLocation ID:
- 2467–2532
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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Accepted Manuscript1.0
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