Negative contact surgery on Legendrian non-simple knot

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Shunyu Wan, University of Virginia
Fine Hall 314

 Etnyre first asked the question on when contact surgery on distinct Legendrian knots produces distinct contact manifolds, and he showed that +1 contact surgeries on certain non Legendrian isotopic representatives of the twist knots always produce the same contact 3-manifold. However, later using linearized contact homology Bourgeois-Ekholm-Eliashberg showed that -1 contact surgery (Legendrian surgery) on those representatives of the twist knots will produce different contact 3-manifolds. Using the contact invariant and Legendrian LOSS invariant in Heegaard Floer theory we are able to show that any contact negative rational surgery (except -1) on the Legendrian representatives of  Legendrian representative of twist knots that have different LOSS invariants produces different contact manifolds with different contact invariants. We will spend most of our time talking about the background on contact geometry and understanding the problem and statement, so everyone is welcome. If there is extra time I will talk about the proof and how Heegaard Floer is involved. This is joint work with Hugo Zhou.