Structures in HOMFLY-PT homology

TL;DR

This work analyzes the structure of triply graded HOMFLY-PT (KR) homology using data from NS and the $\mathfrak{sl}(2)$ action to extract the $S$-invariant and relate it to $\mathfrak{sl}(N)$ invariants. It develops a Delta-grading visualization, studies Rasmussen spectral sequences and their differentials $d_N$, and introduces $d_{a|b}$ and a comprehensive $\mathfrak{sl}(2)$-module framework, including $y$-ification, to organize and interpret the data. The paper proves that for knots in the NS dataset the HOMFLY-PT $S$-invariant is determined by $\mathcal{H}(K)$ and clarifies the relation between $\mathcal{H}(K)$ and $H_{\frak{sl}(N)}(K)$, often yielding a collapse or regrading equivalence for $N\ge 3$, with explicit computations for examples like $10_{125}$, $11n_{135}$, and $T(4,5)$. These results provide a practical route to derive slice-genus bounds and coherence between triply graded and $\frak{sl}(N)$ knot invariants, advancing computational and conceptual understanding of knot concordance through 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 $\mathfrak{sl}(2)$ 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 $\mathfrak{sl}(N)$ concordance invariants.

Figures (26)

• Figure 1: Triply graded homology of the knot $K=11n_{80}$. The $q$ and $a$ gradings are plotted, and $t$ is determined by either $q+a+t=-4$ or $q+a+t=-2$. The red circles indicate the homology of the differentials $d_1$ and $d_{-1}$, in particular, the HOMFLY-PT $S$-invariant equals 2.
• Figure 2: Triply graded homology of the knot $K=11a_{85}$. Since $K$ is two-bridge, it follows that $K$ is thin. The $q$ and $a$ gradings are plotted, and $t$ is determined by $q+a+t=2$. The $S$-invariant equals $-2$.
• Figure 3: Triply graded homology of the knot $K=11a_{263}$. This is the only alternating knot in the dataset which is not thin. The $S$-invariant equals $-8$.
• Figure 4: Here we have both the triply graded homology and $\mathfrak{sl}(3)$ homology of the knot $K=9_{11}$. The $S$-invariant equals $4$ while the $\mathfrak{sl}(3)$ invariants are equal to $j_{x^3-x}=-8$, $s_{x^3-x}=\frac{j_{x^3-x}}{4}=-2$.
• Figure 5: Triply graded homology of the knot $K=\textnormal{$9_{42}$}$. This is an example of a knot whose $S$ invariant, $0$, is not equal to the negative of its signature, $2$.

Theorems & Definitions (73)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Corollary 1.4
• Remark 2.1
• Theorem 2.2: RasTwoBridge
• Remark 2.3
• Proposition 2.4: RasDiff
• Remark 2.5
• Corollary 2.6