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Tanushree Shah

Publications and source records attributed to Tanushree Shah.

10 recordsLinked to original sources
arXiv2026

Contact cosmetic surgery on Legendrian knots in integer homology sphere $L$-spaces

We extend the study of contact cosmetic surgeries to Legendrian knots in integer homology sphere L-spaces . We prove that the contact cosmetic surgery conjecture holds for all non-trivial Legendrian knots in this setting, with the possible exception of Lagrangian slice knots. Our argument adapts and refines techniques from the S3 case to the broader context of L-spaces, incorporating constraints arising from Heegaard Floer theory

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arXiv2026

On Legendrian Thurston-Bennequin-symmetrical graphs

This article reviews the development of Legendrian graph theory in the standard contact 3-sphere ($S^3, \xi_{std}$). We provide a generalized criterion under which the total Thurston-Bennequin invariant of a Legendrian graph (sum of tb of all cycles of the Legendrian graph) can be computed from the tb of its smaller cycles. We verify this criterion for graphs with up to 9 vertices and construct infinite families of examples where it holds. We also present examples demonstrating that each condition in the criterion is necessary. Notably, the graphs satisfying this criterion exhibit a high degree of symmetry.

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arXiv2026

A note on alternating knots in handlebodies

We establish a Kauffman-Murasugi-Thistlethwaite-type theorem for alternating knots in a solid torus. Specifically, we show that any dotted-reduced alternating diagram of a knot in a handlebody realizes the minimal crossing number, and that any two such diagrams of the same knot have identical writhe. The proof relies on a generalization of the Jones polynomial to the setting of handlebodies. A stronger version of this result was already proved by Boden, Karimi, and Sikora using a different generalized Jones polynomial; therefore, this text largely expands on one of the main proof tools.

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arXiv2025

Arithmetic invariants of torus links

The classical analogy between knots and primes motivates the study of Alexander polynomials through an arithmetic perspective. In this article we study the two-parameter family of torus knots and links $T_{p,q}$ and analyze the asymptotic behaviour of the zeros of their Alexander polynomials $\Delta_{p,q}(t)$, defined with respect to the total linking number covering. We prove that as $p,q\to\infty$ these zeros become equidistributed on the unit circle and derive an explicit formula for the limiting frequency with which primitive $r$-th roots of unity appear. To capture finer statistical information, we introduce the moment sequence of the zero distribution and compute its generating function in closed form. We further examine the Iwasawa theory of the corresponding branched covers, determining the Iwasawa invariants. The logarithmic Mahler measure of $\Delta_{p,q}(t)$ vanishes identically and the associated homological growth in towers of abelian covers of $S^3$ branched along $T_{p,q}$ is subexponential.

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arXiv2025

Mixed tori in contact surgery diagrams

We develop a diagrammatic framework for applying the symplectic JSJ decomposition to exact/weak symplectic fillings of 3-dimensional contact manifolds. Namely, we apply the symplectic JSJ decomposition to a contact surgery diagram for some $(Y,\zeta)$, producing a finite collection of contact manifolds, also described diagrammatically, whose exact/weak symplectic fillings determine those of $(Y,\zeta)$. We apply this technique to recover known symplectic filling classifications for certain lens spaces and torus bundles, and also to provide an algorithm for classifying the exact/weak symplectic fillings of a large class of plumbed 3-manifolds.

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arXiv2025

Non-Simple knots in Contact 3-Manifolds

We present new families of examples of non-simple prime Legendrian and transversal knots in tight Lens spaces, which demonstrate that the botany of Legendrians in Lens space is rich. In fact, there are more non-isotopic Legendrians that are topologically isotopic to the $n$-twist knot in a Lens space $L(\alpha, \beta)$ than in $S^3$. We also include connect sum formulas for rational variants of classical invariants, $\mathrm{tb}_\mathbb{Q}$, $\mathrm{rot}_\mathbb{Q}$, and $\mathrm{sl}_\mathbb{Q}$, which indicate that prime knots are the right playground to look for exotic behaviour.

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arXiv2024

Tight contact structures on toroidal plumbed 3-manifolds

We consider tight contact structures on plumbed 3-manifolds with no bad vertices. We discuss how one can count the number of tight contact structures with zero Giroux torsion on such 3-manifolds and explore conditions under which Giroux torsion can be added to these tight contact structures without making them overtwisted. We give an explicit algorithm to construct stein diagrams corresponding to tight structures without Giroux torsion. We focus mainly on plumbed 3-manifolds whose vertices have valence at most 3 and then briefly consider the situation for plumbed 3-manifolds with vertices of higher valence.

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arXiv2024

On contact cosmetic surgery

We demonstrate that the contact cosmetic surgery conjecture holds true for all non-trivial Legendrian knots, with the possible exception of Lagrangian slice knots. We also discuss the contact cosmetic surgeries on Legendrian unknots and make the surprising observation that there are some Legendrian unknots that have a contact surgery with no cosmetic pair, while all other contact surgeries are contactomorphic to infinitely many other contact surgeries on the knot.

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arXiv2023

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 $(\mathbb{S}^3,\xi_{std})$ 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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