The Superpolynomial for Knot Homologies

TL;DR

This work outlines a conjectural triply graded knot homology that categorifies the HOMFLY polynomial and supports a family of differentials {d_N} connecting sl(N) Khovanov–Rozansky homology to knot Floer homology. While a full definition is not given, the authors develop a coherent axiomatic framework, provide numerous consistency checks, and predict how the reduction to KhR_N and HFK should behave for various knots, notably torus knots. The proposal ties together diverse knot invariants via a unified structure, offering concrete predictions for small knots, two-bridge/thin cases, and stable torus knot behavior, with a geometric open Gromov–Witten interpretation offering deeper intuition. If validated, this framework would illuminate the relationships among the HOMFLY, sl(N) homologies, and knot Floer homology, and yield new invariants (such as S) and differential-induced pairings with potential four-ball genus implications.

Abstract

We propose a framework for unifying the sl(N) Khovanov-Rozansky homology (for all N) with the knot Floer homology. We argue that this unification should be accomplished by a triply graded homology theory which categorifies the HOMFLY polynomial. Moreover, this theory should have an additional formal structure of a family of differentials. Roughly speaking, the triply graded theory by itself captures the large N behavior of the sl(N) homology, and differentials capture non-stable behavior for small N, including knot Floer homology. The differentials themselves should come from another variant of sl(N) homology, namely the deformations of it studied by Gornik, building on work of Lee. While we do not give a mathematical definition of the triply graded theory, the rich formal structure we propose is powerful enough to make many non-trivial predictions about the existing knot homologies that can then be checked directly. We include many examples where we can exhibit a likely candidate for the triply graded theory, and these demonstrate the internal consistency of our axioms. We conclude with a detailed study of torus knots, developing a picture which gives new predictions even for the original sl(2) Khovanov homology.

Figures (14)

• Figure 1.1: Triply-graded theory as a unification of knot homologies.
• Figure 3.1: Non-zero differentials for the trefoil knot. On the left is a fully labeled diagram, and on the right is the more condensed form that we will use from now on. The minimum $a$-grading is $2$.
• Figure 3.1: Differentials for $T_{3,4}$.The bottom row of dots has $a$-grading $6$. The leftmost dot on that row has $q$-grading $-6$, which you can determine by noting that the vertical axis of symmetry corresponds to the line $q=0$.
• Figure 3.2: Trefoil with new homological gradings.
• Figure 5.1: Two possible dot diagrams for the knot $9_{42}$. The left-hand diagram assumes that $\mathcal{H}(9_{42})$ is thin and arrives at a contradiction: $d_1^2 \neq 0$. The right-hand diagram corrects this problem by introducing a pair of additional generators.

Theorems & Definitions (19)

• Conjecture 1.2
• Conjecture 1.3
• Conjecture 3.1
• Proposition 3.5
• proof
• Remark 4.1
• Definition 5.1
• Conjecture 5.2
• Proposition 5.3
• Definition 5.5