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Contact structures and reducible surgeries
- Part of
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- Journal:
- Compositio Mathematica / Volume 152 / Issue 1 / January 2016
- Published online by Cambridge University Press:
- 24 September 2015, pp. 152-186
- Print publication:
- January 2016
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- Article
- Export citation
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We apply results from both contact topology and exceptional surgery theory to study when Legendrian surgery on a knot yields a reducible manifold. As an application, we show that a reducible surgery on a non-cabled positive knot of genus
$g$ must have slope 
$2g-1$, leading to a proof of the cabling conjecture for positive knots of genus 2. Our techniques also produce bounds on the maximum Thurston–Bennequin numbers of cables.
