Tautological classes for (n,n+1) torus knots

We construct an explicit isomorphism between the HOMFLY-PT homology of $(n,n+1)$ torus knots and the direct sum of hook isotypic components of the space of diagonal coinvariants. As a consequence, we compute the action of tautological classes in HOMFLY-PT homology of $(n,n+1)$ torus knots and prove that it extends to an action of the Lie algebra of Hamiltonian vector fields on the plane. We also compute the action of differentials $d_N$ in Rasmussen spectral sequences from HOMLFY-PT to $\mathfrak{gl}(N)$ homology of $(n,n+1)$ torus knots.

💡 Research Summary

The paper establishes a precise algebraic bridge between the triply‑graded HOMFLY‑PT homology of the torus knot T(n,n+1) and the space of diagonal coinvariants DRₙ, specifically its hook‑isotypic components. Earlier work by Gorsky, Hogancamp and others showed that the graded dimensions of HHH(T(n,n+1)) coincide with the q,t‑Catalan numbers, which are the Hilbert series of the sign component DR_{sgn} of DRₙ. However, those results did not provide an explicit isomorphism or a way to identify individual homology classes.

The authors first consider the simpler torus link T(n,n). Using the y‑ified version of HOMFLY‑PT homology, denoted HY, they prove that the quotient HY₀(T(n,n))/(x,y) is naturally isomorphic to DR_{sgn}. More generally, HY(T(n,n))/(x,y) ≅ DR_{hook}, where DR_{hook} denotes the direct sum of all hook‑shaped Sₙ‑isotypic components inside DRₙ. This is Theorem 1.2 (a).

A key technical tool is the braid‑like cobordism from T(n,n) to T(n,n+1) that adds (n−1) positive crossings. Elias‑Krasner’s cobordism map π on ordinary HOMFLY‑PT homology lifts to a y‑ified map π_y : HY(T(n,n)) → HY(T(n,n+1)). The authors show that π_y is surjective and induces, after modding out by the maximal ideals (x) or (x,y), the grading‑preserving isomorphism π′ : DR_{sgn} → HHH₀(T(n,n+1)). Consequently, the full homology HHH(T(n,n+1)) is identified with DR_{hook}. This resolves the long‑standing problem of constructing an explicit isomorphism between the two objects.

Having identified the two sides, the paper turns to the “tautological classes” F_k introduced in earlier work on y‑ified homology. These operators satisfy a hard Lefschetz property for k=1 and generate an sl₂ action together with a dual family E_k = Φ F_k Φ, where Φ is an involution swapping the x‑ and y‑variables. The authors prove (Theorem 1.3) that under the isomorphism with DR_{hook}, the operators F_k and E_k become the elementary differential operators

 F_k = Σ_{i=1}^n x_i^k ∂/∂y_i,  E_k = Σ_{i=1}^n y_i^k ∂/∂x_i.

Thus the abstractly defined tautological classes acquire a concrete description as multiplication‑by‑monomial followed by partial differentiation. Moreover, they are non‑zero precisely for k ≤ n−1, reflecting the combinatorial restriction that only hook‑shaped representations of height ≤ n−1 appear in DRₙ.

The paper also addresses the differentials d_N that appear as the first non‑trivial pages of Rasmussen’s spectral sequences from HOMFLY‑PT homology to sl(N) Khovanov‑Rozansky homology. Using the explicit basis of DR_{hook}, the authors compute d_N explicitly and verify that it commutes with all F_k, as predicted by general theory (Proposition 2.10). The formula for d_N matches the conjectural “operator description” previously proposed for torus knots, thereby confirming the conjecture for the case m = n+1 (Theorem 3.20).

A striking corollary (Corollary 1.4) shows that any homogeneous element of HHH(T(n,n+1)) can be obtained from the distinguished generator δ (the rightmost class in the Rouquier complex) by applying a polynomial in the F_k’s together with exterior products of the d_N’s. In other words, the entire homology is “co‑generated” by the tautological classes and the differentials.

Finally, the authors observe that the operators F_k and E_k generate a Lie algebra isomorphic to the subalgebra 𝔥_{≥2}² of Hamiltonian vector fields on the plane spanned by v_{x^a y^b} with a+b≥2. Corollary 1.5 states that the action of F_k and E_k on HHH(T(n,n+1)) extends to a full action of this Hamiltonian Lie algebra. This leads to Conjecture 1.6, proposing that for any link L, the y‑ified homology HY(L) carries an action of the full Hamiltonian Lie algebra 𝔥₂, and for knots the ordinary HOMFLY‑PT homology carries the subalgebra 𝔥_{≥2}².

The paper is organized as follows: Section 2 reviews Soergel bimodules, Rouquier complexes, the y‑ified deformation, the definition of F_k and E_k, and background on diagonal coinvariants and Haiman’s operator conjecture. Section 3 computes HY(T(n,n)), constructs the cobordism map π_y, proves the isomorphisms with DR_{sgn} and DR_{hook}, and establishes the correspondence of tautological classes. Section 3.3 extends the results to all a‑gradings, proves the action of differentials d_N, and discusses the Hamiltonian Lie algebra action. The authors conclude with remarks on potential generalizations and acknowledgments.

Overall, the work provides a complete, constructive identification of torus‑knot HOMFLY‑PT homology with a well‑studied algebraic object, gives explicit formulas for previously mysterious operators, and opens a new avenue linking link homology with symplectic geometry via Hamiltonian vector fields.