Dehn surgeries and negative-definite four-manifolds

Abstract

Dedicated to José Maria Montesinos on the occasion of his 65th birthday.Abstract. Given a knot K in the three-sphere, we address the question: which Dehn surgeries on K bound negative-definite four-manifolds? We show that the answer depends on a number m(K), which is a smooth concordance invariant. We study the properties of this invariant, and compute it for torus knots.

Symplectic Fillings, Contact Surgeries, and Lagrangian Disks

“…Moving to the (2, 2n + 1)-torus knot K n , recall that it was shown in [40] that the result of smooth r-surgery on K n does not admit a symplectically fillable contact structure for r ∈ [2n − 1, 4n), so contact (r)-surgery on the maximal Thurston-Bennequin invariant representative L n of K n with tb(L n ) = 2n − 1 is not symplectically fillable for r < 2n + 1. So to complete the proof of the proposition we need to see that the contact manifold (M n , ξ n ) coming from contact (2n+1)-surgery on L n is Stein fillable, and the result for r > 2n+1 will follow as in the preceding paragraph.…”

“…We begin by noticing that building on work of Ghiggini, Lisca, and Stipsicz [25] and Owens and Strle [40] in certain cases, one can get symplectically fillable contact structures for sufficiently large r. Additionally, for the (2, 2n + 1)-torus knot with maximal Thurston-Bennequin invariant, contact (r)-surgery is tight for all r > 0, and is symplectically (and Stein) fillable if and only if r ≥ 2n + 1.…”

“…So to complete the proof of the proposition we need to see that the contact manifold (M n , ξ n ) coming from contact (2n+1)-surgery on L n is Stein fillable, and the result for r > 2n+1 will follow as in the preceding paragraph. (In [34,40], it is also shown that the result of smooth 4n-surgery on K n admits Stein fillable contact structures, but they are not necessarily the ones coming from contact (2n + 1)surgery on L n . )…”

On eigenvalues of double branched covers

“…It is proved in [11] that for any r ∈ Q >0 , the manifold S 3 r (3 1 ) bounds a negative definite definite 4-manifold if and only if r ≥ 4. Hence neither Σ(10 124 ) nor Σ(10 132 ) can bound a positive definite 4-manifold.…”

Dehn surgeries and rational homology balls

“…Given a knot K ⊂ S 3 Owens and Strle in [22] introduce the following invariant: m(K) = inf{r ∈ Q >0 | S 3 r (K) bounds a negative definite 4-manifold}. They show that m is a concordance invariant and that it vanishes on negative knots.…”