Contact structures and reducible surgeries (Q2787623)

One of the problems in Dehn surgery on a knot \(K\) in \(S^3\) is to determine the knots which admit reducible surgeries. It is known that if \(0\)-surgery on \(K\) is reducible, i.e. some embedded \(2\)-sphere does not bound a ball, then \(K\) is the unknot. Also, since an oriented \(3\)-manifold is prime, i.e. not a non-trivial connected sum, if and only if it is either irreducible or \(S^1\times S^2\), it follows that \(0\)-surgery on a knot is always prime. Many non-trivial knots have reducible surgeries. If \(K\) is the \((p,q)\)-cable of a knot \(K'\) and \(U\) is the unknot, then \(S^3_{pq}(K)=S^3_{p/q}(U)\#S^3_{q/p}(K')\), where \(S^3_{p/q}(U)\) is a lens space. In [Math. Proc. Camb. Philos. Soc. 99, 89--102 (1986; Zbl 0591.57002)], \textit{F.~Gonzáles-Acuña} and \textit{H.~Short} conjectured that these are the only such examples stating that if Dehn surgery on a non-trivial knot \(K\) is reducible, then \(K=C_{p,q}(K')\) for some \(K'\) and the surgery coefficient is \(pq\). This conjecture is known for torus knots and satellite knots, but is still open for hyperbolic knots.NEWLINENEWLINEIn this paper, the authors apply results from both contact topology and exceptional surgery theory to study when Legendrian surgery on a knot can be reducible. For a Legendrian representative of \(K\), performing Legendrian surgery on \(K\) is topologically Dehn surgery with coefficient \(\mathrm{tb}(K)-1\), where \(\mathrm{tb}(K)\) is the Thurston-Bennequin number. First, the authors show that if \(K\) is a knot in \(S^3\) such that \(S^3_n(K)=L(p,q)\#Y\), where \(n<\overline{\mathrm{tb}}(K)\), then \(p=|n|\) with \(n<-1\), \(L(p,q)\) admits a simply-connected Stein filling with intersection form \(\langle n\rangle=\langle -p\rangle\), and \(Y\) is an irreducible integer homology sphere which admits a contractible Stein filling, where \(\overline{\mathrm{tb}}(K)\) is the maximum Thurston-Bennequin number of any Legendrian representative of \(K\). And if \(S^3_n(K)\) has more than two summands, then \(\overline{\mathrm{tb}}(K)\leq n\leq-\overline{\mathrm{tb}}(\overline K)\), where \(\overline K\) is the mirror of \(K\). By using these properties, they prove that if \(K\) is a knot in \(S^3\) such that \(\overline{\mathrm{tb}}(K)\geq 0\), then any surgery on \(K\) with coefficient less than \(\overline{\mathrm{tb}}(K)\) is irreducible. As an application, the authors show that on a non-cabled positive knot \(K\) a reducible surgery \(S^3_n(K)\) of genus \(g(K)\) must have slope \(n=2g(K)-1\), leading to a proof of the Gonzáles-Short cabling conjecture for positive knots of genus \(2\). Finally, the authors make the conjecture that Legendrian surgery on a knot in the tight contact structure on \(S^3\) is never reducible and prove some special cases.