Structures in HOMFLY-PT Homology

Abstract

We study the structure of triply graded Khovanov-Rozansky homology using both the data recently computed by Nakagane and Sano for knots up to 11 crossings, and the $\mathfrak{s}\mathfrak{l}(2)$ action defined by the second author, Hogancamp and Mellit. In particular, we compute the HOMFLY-PT S-invariant for all knots in the dataset, and compare it to the $\mathfrak{s}\mathfrak{l}(N)$ concordance invariants.

KEYWORDS:

• HOMFLY homology
• spectral sequence
• S-invariant

Acknowledgments

We thank Lukas Lewark, Allison Moore, Jake Rasmussen, Paul Wedrich and Melissa Zhang for helpful discussions and suggestions. The reduced Khovanov homology and $\mathfrak{s}\mathfrak{l}(3)$ homology were computed using Lukas Lewark’s programs [16–18], we thank Lukas for help with running them.

Notes

1 Note that this is very different from [3] and some other sources where the numbers indicate the t-grading.

Additional information

Funding

E. G. was partially supported by the NSF grant DMS-1760329, and by the NSF grant DMS-1928930 while E. G. was in residence at the Simons Laufer Mathematical Sciences Institute (previously known as MSRI) in Berkeley, California, during the Fall 2022 semester.