arXiv subjects

Shane D'Mello

Publications and source records attributed to Shane D'Mello.

7 recordsLinked to original sources

arXiv2026

Real rational knots in the quadric of signature~$(3,2)$

We study the space of real rational curves of low degree in the quadric of signature $(3,2)$ and provides a classificaton of real rational knots and nodal curves. Apart from the classification, we also study the relationship between the real rational knots in the quadric and the real rational knots in $\mathbb{RP}^{3}$. Furthermore, a construction for the representatives of all the real rational knots of degree $\leq 5$ in the quadric is presented.

math.GT↗

arXiv2021

Properties of glued knots

We study the properties of glued knots, a sub-class of real rational knots, that can be constructed by gluing ellipses. We define an invariant called the gluing degree and relate it to various classical properties of knots and classify all knots up to gluing degree 6.

math.GT↗

arXiv2016

Constructing real rational knots by gluing

We show that the problem of constructing a real rational knot of a reasonably low degree can be reduced to an algebraic problem involving the pure braid group: expressing an associated element of the pure braid group in terms of the standard generators of the pure braid group. We also predict the existence of a real rational knot in a degree that is expressed in terms of the edge number of its polygonal representation.

math.GT↗

arXiv2016

M-curves and symmetric products

Let $(X , σ)$ be a geometrically irreducible smooth projective M-curve of genus $g$ defined over the field of real numbers. We prove that the $n$-th symmetric product of $(X , σ)$ is an M-variety for $n=2 ,3$ and $n\geq 2g -1$.

math.AG↗

arXiv2015

Rational cuspidal curves on del-Pezzo surfaces

We obtain an explicit formula for the number of rational cuspidal curves of a given degree on a del-Pezzo surface that pass through an appropriate number of generic points of the surface. This enumerative problem is expressed as an Euler class computation on the moduli space of curves. A topological method is employed in computing the contribution of the degenerate locus to this Euler class.

math.AG↗

arXiv2013

Classification of real rational knots of low degree in the 3-sphere

In this paper we classify, up to rigid isotopy, non-singular real rational curves of degrees less than or equal to 6 in a quadric homeomorphic to the 3-sphere. We also study their connections with rigid isotopy classes of real rational knots in $\mathbb{RP}^3$.

math.GT↗

arXiv2013

Rigid isotopy classification of real degree-4 planar rational curves with only real nodes (An elementary approach)

We prove an elementary method to classify, up to rigid isotopy, all nodal degree 4 real rational curves in $\mathbb{RP}^2$ that have only real double points. We show how to associate a chord diagram to a nodal real degree 4 planar rational curve and prove that such curves are defined up to rigid isotopy by their chord diagrams.

math.AG↗