Spatial knots: how do shapes intertwine in the universe?

Imagine having an infinite string and wanting to intertwine it within a very strange world. Not a normal world, but a space where, if you walk in one direction, you return exactly to your starting point, as if you were on a massive cosmic treadmill. In mathematics, this scenario is called the product of a surface and a circle.

Recently, researchers discovered an incredible way to describe these complex entanglements. Instead of using infinitely complex drawings, they realized we can use 'virtual diagrams' with special lines called 'double lines.' It is a bit like using secret symbols to explain how a thread wraps around itself in a curved space.

This discovery is fundamental because it proves that virtual knot theory is actually a part of a much larger and more fascinating system. In practice, studying these knots allows us to better understand how links between objects behave in curved and complex spaces. It is as if we have found a new magnifying glass to view the geometry of the universe and understand how shapes can fit together without ever breaking or separating!

The research focuses on the topology of knots within three-dimensional manifolds, specifically the product of a genus 'g' surface and a circle ($S_g \times S^1$). The central problem concerns the representation of these knots through diagrams that use 'double lines' to indicate passage through the circular component of the manifold. The authors demonstrate that virtual knot theory is a subcase or is 'embedded' in the theory of knots in these manifolds. Furthermore, the study explores the connection between knots in $S^2 \times S^1$ and two-component links in $S^3$, where one component is a trivial knot. This approach allows for the analysis of the minimality and separability of classical links using tools derived from the topology of more complex manifolds, providing new methods to classify entanglements in non-Euclidean spaces.