The purpose of this research is to construct a new invariant of Legendrian knots in contact 3-manifolds and to investigate its properties.  The study on Legendrian knots plays an important role in 3-dimensional contact topology.  I defined new invariants of Legendrian knots using rack colorings and proved that those invariants can distinguish Legendrian unknots.A paper on the results titled “Bi-Legendrian rack colorings of Legendrian knots” is to appear in Journal of Knot Theory and Its Ramifications.Another theme in this research is a Legendrian embedding of a graph into the 3-space as a generalization of Legendrian knots.  Ayumu Inoue, Ryo Nikkuni, Kouki Taniyama and I proved that for a graph which satisfies some conditions, the parity of the sum of crossing numbers of cycles in the graph immersed into the plane is independent of the choice of the immersion.  As a corollary of the theorem, we showed that the Petersen graph and the Heawood graph have no Legendrian embeddings satisfying a certain condition.A paper on the results titled “Crossing numbers and rotation numbers of cycles in a plane immersed graph” has published in Journal of Knot Theory and Its Ramifications.

A Jacobi structure on a manifold is a generalization of both of a contact structure and a Poisson structure.  Geometric properties of contact manifolds and Poisson manifolds are understood to some extent.  Meanwhile, geometric properties of Jacobi manifolds are hardly understood.  In studying geometric properties of manifolds with some geometric structures, it is often useful to introduce metrics which are compatible in some sense with those structures.Tomoya Nakamura and I defined the compatibility between Jacobi structures and pseudo-Riemannian metrics by using the Levi-Civita connection of the metric.  This compatibility is considered as a generalization of the compatibility between Poisson structures and pseudo-Riemannian metrics defined by Boucetta.  We showed that this compatibility behaves well to the Poissonization of a Jacobi structure.  In addition, we proved that if a contact metric structure is compatible, then it becomes a Sasaki structure.  Hence our definition of the compatibility between Jacobi structures and metrics is regarded as a generalization of Sasaki structures.  We are writing a paper on these results and will submit it to an international academic journal.