Curves In 3-Manifolds

This site is a place to explore beautiful ideas in geometry and topology - discover loops, structures, and visual ways to understand complicated spaces!

This project looks at how loops behave in certain spaces by creating visual representations. In particular, we focus on a space called the Seifert fiberings of $S^2 \times S^1$, which is related to the loop space of the rotation group $SO(3)$. Although algebraic concepts like homology can describe these spaces, they are often hard to understand just by reading formulas or definitions.

To make these ideas more accessible, we use visualizations to show how the basic shapes, or generators, of these spaces look. Instead of using only abstract calculations, we show how these curves appear when placed inside the familiar space $S^2 \times S^1$. Our goal is to give people a more intuitive and concrete way to explore and understand the structure of these loops.

The results displayed on this site are largely outlined in this paper.

More Information

For an introductory explanation of these animations, refer to this article.

Additionally, you can find in this album a collection of all the images we’ve generated over the course of a semester.

You can also find the code to generate some images on your own in this GitHub repository.

Acknowledgments

This project was completed in part with the Illinois Mathematics Lab at the University of Illinois Urbana-Champaign.