Negative contact surgery on Legendrian non-simple knot

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Shunyu Wan, University of Virginia

ÌýEtnyre first asked the question on when contact surgery on distinct Legendrian knotsÌýproduces distinct contact manifolds, andÌýhe showed that +1 contact surgeries on certainÌýnon Legendrian isotopic representatives of the twist knotsÌýalways 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 knotsÌýwill produceÌýdifferent contact 3-manifolds. Using the contact invariant and Legendrian LOSS invariant in Heegaard Floer theoryÌýwe are able to show that any contact negative rational surgery (except -1) on the Legendrian representatives ofÌýÌýLegendrian 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 understandingÌýthe problem and statement, so everyone is welcome. If there is extra time I will talk about theÌýproof and how Heegaard Floer is involved. This is joint work with Hugo Zhou.