HOMFLY-PT Quivers of Rational Knots

This program is used to compute the HOMFLY-PT quiver data for a given rational knot \(K_{u/v}\). From this data, we can compute the colored HOMFLY-PT polynomials.

Authors:

Elijah Berry · Eduardo Perez · Henry Zaslow · Rongjiang Guo

About

This code was written for the UIUC Summer IML project: Colored HOMFLY-PT polynomials of Rational Knots. We developed a fast algorithm to compute colored HOMFLY-PT polynomials, which then can be specialized to the colored Jones and colored \(\mathfrak{sl}_N\) polynomials. We have also implemented functions to visualize these polynomials, and how they vary with color. In particular, we visualize how tails and heads form for each of these polynomials. These polynomials are calculated by means of the Knots-Quivers Correspondence for rational knots. Thus the computation of these polynomials is reduced to calculating the quiver data of the knots. Higgins details a geometric method to calculate these quivers based on winding numbers of curves in the plane, of which we make use in our algorithm.

Visualizations

Figure 1. The 5th-Colored HOMFLY-PT Polynomial of \( K_{5/2}. \)

Figure 2. The 10th-Colored HOMFLY-PT Polynomial of \( K_{5/2}. \)

Figure 3. The Tail of the Colored HOMFLY-PT Polynomial of \( K_{5/2}. \)

Acknowledgements

This project was fully supported by NSF Grant DMS-2405302. Additionally, we would like to thank Jonathan Higgins and Jake Rasmussen for leading this project and for their support.

Pages

• Read the tutorial
• Github Repository
• API Documentation
• Gallery
• Quiver Tables

Contact

If you have questions about the project, encounter an issue, or would like to learn more about the software, please contact us at [email protected], [email protected], [email protected], and [email protected].