What is this repository for?

This repository supplements the paper

L. Lewark and C. Zibrowius, Rasmussen invariants of Whitehead doubles and other satellites

For any c equal to a prime number or 0, the paper defines certain knot invariants ϑ_c that govern the behaviour of Rasmussen invariants over fields of characteristic c under satellite operations with patterns of winding number 0 and wrapping number 2.

We collect all known ϑ_c-invariants in the following table (index.html):

https://llewark.github.io/theta/.

What are the other files for?

The table is generated from the source file data.csv by running the script converter.py:

./converter.py data.csv > index.html

Each line of data.csv corresponds to one of the following:

• a computation, in which case the format is a tab separated list of
  • the knot id, e.g. 3_1 for the trefoil knot,
  • the invariant, e.g. theta_2 for ϑ₂,
  • the value of the invariant, e.g. 4,
  • metadata, which is formatted as a semi-colon (;) separated list of any number of
    • colon (:) separated key-value pairs, e.g. program:khoca and
    • comments, e.g. My computer was very tired after this computation.
• a comment about a particular knot, in which case the format is a tab separated list of
  • the knot id and
  • a comment.

The script performs some basic sanity checks along the way. For example, it ensures that if multiple computations were made for the same invariant, the results all agree.

Where does the data come from?

The values for the ϑ_c-invariants were computed using the following two programs:

• khoca, a program for computing sl(N)-homology theories of knots, written by L. Lewark.
  Most computations were done using this program.

• kht++, a program for calculating the Khovanov and Bar-Natan homology of links and tangles, written by C. Zibrowius.
  Details for computations done with this program are available here.

The values of some other invariants, such as the τ invariant and the slice genus, were scraped from knotinfo:

C. Livingston and A. H. Moore, KnotInfo: Table of Knot Invariants

For the source of any value, please check the metadata by clicking on the corresponding row of the table.

Acknowledgments

While working on this repository in 2022, Lukas Lewark and Claudius Zibrowius were supported by the Emmy Noether Programme of the DFG, project number 412851057.