Georgia Institute of Technology College of Sciences

The existence and nonexistence of tight contact structures on the 3-manifold are interesting and important topics studied over the past thirty years. Etnyre-Honda found the first example of a 3-manifold that does not admit tight contact structure, and later Lisca-Stipsicz extended their result and showed that a Seifert fiber space admits a tight contact structure if and only if it is not the smooth (2n − 1)-surgery along the T(2,2n+1) torus knot for any positive integer n.

Surprisingly, since then no other example of a 3-manifold without tight contact structure has been found. Hence, it is interesting to study if all such manifolds, except those mentioned above, admit a tight contact structure. Towards this goal, I will discuss the joint work with Zhenkun Li and Hugo Zhou about showing any negative surgeries on any knot in S^3 admit a tight contact structure.