arXiv subjects

Prerak Deep

Publications and source records attributed to Prerak Deep.

4 recordsLinked to original sources

arXiv2026

Contact Surgery Numbers of the 3-torus

We study contact surgery numbers for contact structures on the 3-torus. We show that all contact structures obtained via contact surgery along a Legendrian Borromean ring are overtwisted, and that this construction yields infinitely many pairwise non-contactomorphic contact structures on $\mathbb{T}^3$. We prove an obstruction on the maximal Thurston-Bennequin invariant of 3-component links to produce $\mathbb{T}^3$ by Dehn surgery. On a side note, using Legendrian surgery and the ruling invariant, we show that the total symplectic homology of Stein fillings of $(\mathbb{T}^3, \xi_1)$ is non-zero.

math.SG↗

arXiv2025

On Contact Round Surgeries on $(\mathbb{S}^3,\xi_{st})$ and Their Diagrams

We introduce the notion of contact round surgery of index $1$ on Legendrian knots in a general contact 3-manifold. It generalizes the notion of contact round surgery of index 1 on Legendrian knots introduced by Adachi. In $\left(\mathbb{S}^3, \xi_{st}\right)$, we introduce the notion of contact round surgery of index 2 on a Legendrian knot and realize Adachi's contact round 2-surgery on a convex torus as a contact round surgery of index $2$ on a Legendrian knot in $\left(\s^3, \xi_{st}\right)$. We associate surgery diagrams to contact round surgeries of indices 1 and 2 on Legendrian knots in $\left(\mathbb{S}^3, \xi_{st}\right)$. With this set-up, we show that every closed connected contact 3-manifold can be obtained by performing a sequence of contact round surgeries on some Legendrian link in $\left(\mathbb{S}^3, \xi_{st}\right)$, thus obtaining a contact round surgery diagram for each contact 3-manifold. This is analogous to the result of Ding-Geiges for contact Dehn surgeries. We also discuss a bridge between certain pairs of contact round surgery diagrams of indices 1 and 2, and contact $(\pm1)$-surgery diagrams. We use this bridge to establish the result mentioned above. In the end, we derive a corollary that gives sufficient conditions on contact round surgeries to produce symplectically fillable manifolds.

math.SG↗

arXiv2025

On Round Surgery Diagrams For 3-Manifolds

We introduce the notion of round surgery diagrams in $S^3$ for representing 3-manifolds similar to Dehn surgery diagrams. We give a correspondence between a certain class of round surgery diagrams and Dehn surgery diagrams for 3-manifolds. As a consequence, we recover Asimov's result, stating that any closed connected oriented 3-manifold can be obtained by a round surgery on a framed link in $S^3$. There may be more than one round surgery diagram giving rise to the same 3-manifold. Thus, it is natural to ask whether there is a version of Kirby Calculus for round surgery diagrams, similar to the case of Dehn surgery diagrams with integral framings. In this direction, we define four types of moves on round surgery diagrams such that any two round surgery diagrams corresponding to the same 3-manifold can be obtained one from another by a finite sequence of these moves, thereby establishing a version of Kirby Calculus. As an application, we prove the existence of taut foliations, hence the existence of tight contact structures on 3-manifolds obtained by round 1-surgery on fibred links with two components on $S^3$.

math.GT↗

arXiv2024

On A Potential Contact Analogue Of Kirby Move Of Type 1

In this expository note, we explore the possibility of the existence of Kirby move of type 1 for contact surgery diagrams. In particular, we give the necessary conditions on a contact surgery diagram to become a potential candidate for contact Kirby move of type 1. We observe that there is a collection of contact positive integral surgery diagrams on Legendrian unknots satisfying those conditions.

math.GT↗