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2 results

  • On the contact mapping class group of Legendrian circle bundles

  • Part of
  • Emmanuel Giroux, Patrick Massot
  • Journal:
    Compositio Mathematica / Volume 153 / Issue 2 / February 2017
    Published online by Cambridge University Press:
    01 February 2017, pp. 294-312
    Print publication:
    February 2017
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  • In this paper, we determine the group of contact transformations modulo contact isotopies for Legendrian circle bundles over closed surfaces of non-positive Euler characteristic. These results extend and correct those presented by the first author in a former work. The main ingredient we use is connectedness of certain spaces of embeddings of surfaces into contact 3-manifolds. This connectedness question is also studied for itself with a number of (hopefully instructive) examples.

  • Contact structures and reducible surgeries

  • Part of
  • Tye Lidman, Steven Sivek
  • 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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  • 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.