Contact surgery numbers

It is known that any contact $3$-manifold can be obtained by rational contact Dehn surgery along a Legendrian link $L$ in the standard tight contact $3$-sphere. We define and study various versions of contact surgery numbers, the minimal number of components of a surgery link $L$ describing a given contact $3$-manifold under consideration.

In the first part of the paper, we relate contact surgery numbers to other invariants in terms of various inequalities. In particular, we show that the contact surgery number of a contact manifold is bounded from above by the topological surgery number of the underlying topological manifold plus three.

In the second part, we compute contact surgery numbers of all contact structures on the $3$-sphere. Moreover, we completely classify the contact structures with contact surgery number one on $S^1 \times S^2$, the Poincaré homology sphere and the Brieskorn sphere $\Sigma(2,3,7)$.We conclude that there exist infinitely many non-isotopic contact structures on each of the above manifolds which cannot be obtained by a single rational contact surgery from the standard tight contact $3$-sphere. We further obtain results for the $3$-torus and lens spaces.

As one ingredient of the proofs of the above results we generalize computations of the homotopical invariants of contact structures to contact surgeries with more general surgery coefficients which might be of independent interest.

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Received 22 March 2022

Received revised 16 April 2023

Accepted 13 May 2023

Published 6 June 2024