arXiv subjects

Visakh Narayanan

Publications and source records attributed to Visakh Narayanan.

4 recordsLinked to original sources

arXiv2025

Knots and non-orientable surfaces in 3-manifolds

In this article, we propose a new approach for describing and understanding knots and links in a 3-manifold through the use of an embedded non-orientable surface. Specifically, we define a plat-like representation based on this non-orientable surface. The method applies to manifolds of the form $M=\mathcal H\cup_φ \mathcal C(U)$ where $\mathcal H$ is a handlebody, $\mathcal C(U)$ is the mapping cylinder of the orientating two sheeted covering of a non-orientable closed surface $U$ and $φ:\partial \mathcal H\to \partial \mathcal C(U)$ is an attaching homeomorphism. We show that, by fixing such a splitting any link in the manifold can be represented as a plat-like closure of an element of the surface braid group of $\partial \mathcal H$. Manifolds of this type were extensively studied by J.H. Rubinstein \cite{rubinstein1978one}, where it is shown that any 3-manifold $M$, with a non-vanishing $H_2(M,\frac{\mathbb{Z}}{2\mathbb{Z}})$ will admit such a splitting. Thus the method is quite general. We provide explicit examples of such embeddings in lens spaces $L(2k,q)$ and the trivial circle bundles over orientable closed surfaces, $Σ\times S^1$

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arXiv2024

Knots in $\mathbb{R}P^3$

This paper studies knots in three dimensional projective space. Our technique is to associate a virtual link to a link in projective space so that equivalent projective links go to equivalent virtual links (modulo a special flype move). We apply techniques in virtual knot theory to obtain a Jones polynomial for projective links. We show that this is equivalent to the known Jones polynomial defined by Drobotukhina for them. We apply virtual Khovanov homology and the virtual Rasmussen invariant of Dye, Kaestner, and Kauffman to projective links. We compare this cohomology theory with the Khovanov type theory developed by Manolescu and Willis for projective knots. We show that these theories are essentially equivalent.

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arXiv2023

Plat closures of spherical braids in $\mathbb{R}P^3$

We define plat closure for spherical braids to obtain links in $\mathbb{R}P^3$ and prove that all links in $\mathbb{R}P^3$ can be realized in this manner. Given a spherical braid $β$ of $2n$ strands in $\mathbb{R}P^3$ we associate a permutation $h_β$ on $n$ elements called \textit{residual permutation}. We prove that the number of components of the plat closure link of a spherical braid $β$ is same as the number of disjoint cycles in $h_β$. We also present a set of moves on spherical braids in the same spirit as the classical Markov moves on braids. The completeness of this set of moves to capture the entire isotopy classes of the plat closure links is still to be explored.

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arXiv2023

Geometry of knots in real projective $3$-space

This paper discusses some geometric ideas associated with knots in real projective 3-space $\mathbb{R}P^3$. These ideas are borrowed from classical knot theory. Since knots in $\mathbb{R}P^3$ are classified into three disjoint classes, - affine, class-$0$ non-affine and class-$1$ knots, it is natural to wonder in which class a given knot belongs to. In this paper we attempt to answer this question. We provide a structure theorem for these knots which helps in describing their behaviour near the projective plane at infinity. We propose a procedure called {\it space bending surgery}, on affine knots to produce several examples of knots. We later show that this operation can be extended on an arbitrary knot in $\mathbb{R}P^3$. We also define a notion of \say{ genus} for knots in $\mathbb{R}P^3$ and study some of its properties. We prove that this genus detects knottedness in $\mathbb{R}P^3$ and gives some criteria for a knot to be affine and of class-$1$. We also prove a \say{non-cancellation} theorem for space bending surgery using the properties of genus. We produce examples of class-$0 $ non-affine knots with genus $1$. And finally we study the notion of companionship of knots in $\mathbb{R}P^3$ and using that we provide a geometric criteria for a knot to be affine. Thus we highlight that, $\mathbb{R}P^3$ admits a knot theory with a truly different flavour than that of $S^3$ or $\mathbb{R}^3$.

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