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Dheeraj Kulkarni

Publications and source records attributed to Dheeraj Kulkarni.

15 recordsLinked to original sources
arXiv2026

On Round Surgery on Framed Links and Their Diagrams for 3-manifolds

We introduce a novel method for performing round surgery on framed links in a 3-manifold. In $\mathbb{S}^3$, these operations give rise to round surgery diagrams for 3-manifolds, directly analogous to classical Dehn surgery diagrams. We establish an explicit correspondence between a specific class of these round surgery diagrams and integral Dehn surgery diagrams. As a consequence, we prove a round surgery analogue of the Lickorish-Wallace theorem, demonstrating that any closed, connected, oriented 3-manifold can be obtained via round surgery on a framed link in $\mathbb{S}^3$. This result recovers Asimov's foundational theorem for oriented 3-manifolds within a purely diagrammatic framework. Since distinct round surgery diagrams can represent homeomorphic 3-manifolds, we naturally investigate whether a version of Kirby calculus exists in this setting. To this end, we define four types of diagrammatic moves and prove that any two round surgery diagrams representing the same 3-manifold are related by a finite sequence of these moves, thereby establishing a "round version" of Kirby calculus. Finally, we prove two applications of this framework. First, we outline a method to construct standard Kirby diagrams directly from round surgery diagrams. Second, we establish the existence of taut foliations-and consequently tight contact structures-on 3-manifolds obtained via round surgeries of index 1 on two-component fibered links in $\mathbb{S}^3$.

math.GT↗
arXiv2025

On Contact Round Surgeries on $(\mathbb{S}^3,ξ_{st})$ 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 $\left(\mathbb{S}^3, ξ_{st}\right)$, 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 $\left(\s^3, ξ_{st}\right)$. We associate surgery diagrams to contact round surgeries of indices 1 and 2 on Legendrian knots in $\left(\mathbb{S}^3, ξ_{st}\right)$. 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 $\left(\mathbb{S}^3, ξ_{st}\right)$, 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 The Cost Function Associated With Legendrian Knots

In this article, we introduce a non-negative integer-valued function that measures the obstruction for converting topological isotopy between two Legendrian knots into a Legendrian isotopy. We refer to this function as the Cost function. We show that the Cost function induces a metric on the set of topologically isotopic Legendrian knots. Hence, the set of topologically isotopic Legendrian knots can be seen as a graph with path-metric given by the Cost function. Legendrian simple knot types are shown to be characterized using the Cost function. We also get a quantitative version of Fuchs-Tabachnikov's Theorem that says any two Legendrian knots in $(\mathbb{S}^3,ξ_{std})$ in the same topological knot type become Legendrian isotopic after sufficiently many stabilizations. We compute the Cost function for Legendrian simple knots (for example torus knots) and we note the behavior of Cost function for twist knots and cables of torus knots (some of which are Legendrian non-simple). We also construct examples of Legendrian representatives of 2-bridge knots and compute the Cost between them. Further, we investigate the behavior of the Cost function under the connect sum operation. We conclude with some questions about the Cost function, its relation with the standard contact structure, and the topological knot type.

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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↗
arXiv2023

On Certain Rigidity Results of Compact Regular $(κ, μ) $-Manifolds

In this article, we investigate the Riemannian and semi-Riemannian metrics on the base space of the Boothby-Wang fibration of a closed regular non-Sasakian $(κ, μ)$-manifold. To this end, we study a natural class of deviations of the projection map from being (semi-)Riemannian submersions. We consider deviations that preserve the canonical bi-Legendrian structure on the given $(κ, μ)$-manifold. We present rigidity results for Riemannian and semi-Riemannian metrics on the base space which orthogonalize the natural bi-Lagrangian structure induced by the $(κ, μ)$-structure. This approach gives a unified framework to analyze rigidity results in both categories. More precisely, in the Riemannian category, we obtain uniqueness of Sasakian structure on the given $(κ, μ)$-manifold which orthogonalizes the canonical bi-Legendrian structure. In the semi-Riemannian category, we obtain an explicit description of the finitely many para-contact structures which orthogonalize the canonical bi-Legendrian structure.

math.DG↗
arXiv2023

Topological Fundamental Groupoid. II. An action category of the fundamental groupoid

For a path connected, locally path connected and semilocally simply connected space $X$, let $Π_1(X)$ denote its topologised fundamental groupoid as established in the first article of this series. Let $\mathcal{E}$ be the category of $Π_1(X)$-spaces in which the momentum maps are local homeomorphisms. We show that this category is isomorphic to that of covering spaces of $X$. Using this, we give different characterisations for free or proper actions of the fundamental groupoid in $\mathcal{E}$.

math.AT↗
arXiv2022

On a generalization of Jones polynomial and its categorification for Legendrian Knots

In this article, we explore a polynomial invariant for Legendrian knots which is a natural extension of Jones polynomial for (topological) knots. To this end, a new type of skein relation is introduced for the front projections of Legendrian knots. Further, we give a categorification of the polynomial invariant for Legendrian knots which is a natural extension of Khovanov homology for knots. The Thurston-Bennequin invariant of Legendrian knot appears naturally in the construction of the homology as the grade-shift. The constructions of the polynomial invariant and its categorification are natural in the sense that if we treat Legendrian knots as only knots (that is, we forget the geometry on the knots), then we recover the Jones polynomial and Khovanov homology respectively. In the end, we discuss strengths and limitations of these invariants.

math.GT↗
arXiv2020

Periodic Surface Homeomorphisms and Contact Structures

Periodic surface homemorphisms (diffeomorphisms) play a significant role in the the Nielsen-Thurston classification of surface homeomorphisms. Periodic surface homeomorphisms can be described (up to conjugacy) by using data sets which are combinatorial objects. In this article, we start by associating a rational open book to a slight modification of a given data set, called marked data set. It is known that every rational open book supports a contact structure. Thus, we can associate a contact structure to a periodic map and study the properties of it in terms combinatorial conditions on marked data sets. In particular, we prove that a class of data sets, satisfying easy-to-check combinatorial hypothesis, gives rise to Stein fillable contact structures. In addition to the above, we prove an analogue of Mori's construction of explicit symplectic filling for rational open books. We also prove a sufficient condition for Stein fillability of rational open books analogous to the positivity of monodromy in honest open books as in the result of Giroux and Loi-Piergallini.

math.GT↗
arXiv2019

Optimal Windowing of MR Images using Deep Learning: An Enabler for Enhanced Visualization

Window width (WW) and window level (WL) adjustments aid in visualizing anatomies with a suitable contrast. However, the presence of background noise in MR images biases the calculation of default WW/WL values since it necessitates a trade-off between enhancing contrast of foreground/anatomy of interest vs suppressing background/ outside the anatomy of interest. This paper proposes an intelligent algorithm to improve the automatic computation of WW/WL and provide better control for user defined windowing.This is achieved by first eliminating the background pixels using a Deep Neural network and then computing WW/WL.

eess.IV↗
arXiv2018

On the existence of non-trivial laminations in $\mathbb{CP}^2$

In this article, we show the existence of a nontrivial Riemann surface lamination embedded in $\mathbb{CP}^2$ by using Donaldson's construction of asymptotically holomorphic submanifolds. Further, the lamination we obtain has the property that each leaf is a totally geodesic submanifold of $\mathbb{CP}^2 $ with respect to the Fubini-Study metric. This may constitute a step in understanding the conjecture on the existence of minimal exceptional sets in $\mathbb{CP}^2$.

math.DG↗
arXiv2018

A Compactness Theorem for Embedded Measured Riemann Surface Laminations

We prove a compactness theorem for embedded measured hyperbolic Riemann surface laminations in a compact almost complex manifold $(X, J)$. To prove compactness result, we show that there is a suitable topology on the space of measured Riemann surface laminations induced by Levy-Prokhorov metric. As an application of the compactness theorem, we show that given a biholomorphism of $ϕ$ of a closed complex manifold $X$, some power $ϕ^k $ ($k>0$) fixes a measured Riemann surface lamination in $X$.

math.GT↗
arXiv2017

On Rack Invariants Of Legendrian Knots

In this article, we introduce rack invariants of oriented Legendrian knots in the 3-dimensional Euclidean space endowed with the standard contact structure, which we call Legendrian racks. These invariants form a generalization of the quandle invariants of knots. These rack invariants do not result in a complete invariant, but detect some of the geometric properties such as cusps in a Legendrian knot. In the case of topologically trivial Legendrian knots, we test this family of invariants for its strengths and limitations. We further prove that these invariants form a natural generalization of the quandle invariant, by which we mean that any rack invariant under certain restrictions is equivalent to a Legendrian rack. The axioms of these racks are expressible in first order logic, and were discovered through a series of experiments using an automated theorem prover for first order logic. We also present the results from the experiments on Legendrian unknots involving auto-mated theorem provers, and describe how they led to our current formulation.

math.GT↗
arXiv2016

Minimal contact triangulations of 3-manifolds

In this paper, we explore minimal contact triangulations on contact 3-manifolds. We give many explicit examples of contact triangulations that are close to minimal ones. The main results of this article say that on any closed oriented 3-manifold the number of vertices for minimal contact triangulations for overtwisted contact structures grows at most linearly with respect to the relative $d^3$ invariant. We conjecture that this bound is optimal. We also discuss, in great details, contact triangulations for a certain family of overtwisted contact structures on 3-torus.

math.GT↗