Legendrian Reidemeister moves for the convex surface projection
We prove a Legendrian Reidemeister theorem for Legendrian knots in thickened convex surfaces.
math.SG↗
arXiv subjects
Zijian Rong
Publications and source records attributed to Zijian Rong.
Invariants of Legendrian knots in thickened convex surfaces
We define a differential graded algebra associated to Legendrian knots in thickened convex surfaces $Σ\times \mathbb{R}$. The algebra is defined in the same spirit as the Chekanov-Eliashberg DGA for Legendrians in $\mathbb{R}^3$, but makes use of the data of the dividing set $Γ$ of $Σ$. The algebra is generated by countably many Reeb chords of the Legendrian $Λ$, and its differential counts certain immersed polygons in the projection $π:Σ\times \mathbb{R}\to Σ\times \{0\}$ with boundary on $π(Λ)\cup Γ$. We show that the differential squares to zero and that the stable tame isomorphism type of the DGA is invariant under Legendrian isotopy. Finally, we compute several examples and use the invariant to distinguish Legendrian knots in thickened convex surfaces that cannot be distinguished by the classical invariants.
math.SG↗
Filtering cohomology of ordinary and Lagrangian Grassmannians
This paper studies, for a positive integer $m$, the subalgebra of the cohomology ring of the complex Grassmannians generated by the elements of degree at most $m$. We build in two ways upon a conjecture for the Hilbert series of this subalgebra due to Reiner and Tudose. The first reinterprets it in terms of the operation of $k$-conjugation, suggesting two conjectural bases for the subalgebras that would imply their conjecture. The second introduces an analogous conjecture for the cohomology of Lagrangian Grassmannians.
math.CO↗