On Knotted Spheres in Euclidean $4$-space $ \mathbb{E}^{4}$

TL;DR

This paper investigates the geometric properties of knotted spheres in four-dimensional Euclidean space, focusing on curvature, conjugate nets, and Laplace transforms to deepen understanding of their structure.

Contribution

It provides new insights into the curvature and transformation properties of knotted spheres in 4D space, expanding the mathematical understanding of these complex surfaces.

Findings

Derived basic curvature properties of knotted spheres

Analyzed conjugate nets on these surfaces

Explored Laplace transforms related to knotted spheres

Abstract

In the present study we consider knotted spheres in Euclidean $4$-space $ \mathbb{E}^{4}$. Firstly, we give some basic curvature properties of knotted spheres in $ \mathbb{E}^{4}$. Further, we obtained some results related with the conjugate nets and Laplace transforms of these kind of surfaces.