arXiv subjects
Publications and source records attributed to Matthias Scharitzer.
In the appendix of [LSV26], the authors produce an invariant for contact structures on $\Sigma \times [0,1]$. In this paper, we prove a result announced there which states that this is in-fact a full invariant. As an application, we provide an independent proof of Eliashbergs theorem and show that various construction methods for contact structures are equivalent.
We show that the $q$-dilogarithm identities associated to the unidirectional $A_n$-quiver lift to the HOMFLYPT-skein algebra of a genus $n$ handlebody.
This paper proves the Giroux Correspondence in dimension three using Heegaard splittings of contact manifolds. In two of the authors earlier paper they proved the Giroux Correspondence for tight contact 3-manifolds via convex Heegaard surfaces, and simultaneously, Honda, Breen and Huang gave an alldimensions proof of the Giroux Correspondence by generalising convex surface theory to higher dimensions. This paper extends the Heegaard splitting approach to arbitrary (not necessarily tight) contact 3-manifolds in order to provide a proof accessible to a low-dimensional audience. The proof assumes classification moves relating bypass decompositions for isotopic contact structures on cobordisms that are topological products; in the Appendix, we prove this result in the 3- dimensional setting.
Under certain hypotheses, we show that Legendrian surfaces related by disk surgery will have q-deformed augmentation spaces that are related by q-deformed cluster transformation. The proof is geometric, via considerations of moduli of holomorphic curves. In fact, our results naturally give a more general HOMFLYPT "skein-valued cluster transformation", of which the q-cluster transformation is the U(1) specialization. We apply our methods to the Legendrians associated to cubic planar graphs, where mutation of graphs lifts to Legendrian disk surgery. We show that their skein-valued mirrors transform by skein-valued cluster transformation, and give a formula for the skein-valued curve counts on their fillings.
For a certain class of Legendrian surfaces in the five-sphere, associated to cubic planar graphs, we show that the all-genus skein-valued holomorphic curve invariants of any filling are annihilated by certain explicit skein-valued operator equations.