Link homology and loop homology

Abstract: We compute the $k$-colored $\mathfrak{sl}(N)$ homology of the torus knot $T(2,2m+1)$, and we show that it stabilizes as $m\to\infty$ to the integral homology of the free loop space of the complex Grassmannian $\mathrm{Gr}(k,N)$. In particular, when $k = 1$ and $N = 2$, we observe that the Khovanov homology of $T(2,2m+1)$ stabilizes to the homology of the free loop space of the $2$-sphere.

• The paper proves that stable k-colored sl(N) Khovanov–Rozansky homology of T(2,∞) matches the integral homology of the free loop space LGr(k,N), including partition-indexed grading shifts.
• The geometric computation uses Morse–Bott theory for closed geodesics on Grassmannians, while the algebraic computation uses minimal Rickard complexes and a nil-Hecke action by chain maps.
• The result extends colored torus-knot calculations beyond the trefoil but remains computational, with the conceptual link to loop spaces and the role of the resulting cyclic Z/2N grading left open.

Overview

This paper, by Joshua Wang (2604.25174), establishes an isomorphism between two a priori unrelated objects: the stable $k$-colored $sl(N)$ Khovanov–Rozansky homology of the torus link $T(2,\infty)$ and the integral homology of the free loop space of the complex Grassmannian $\Gr(k,N)$. The result rests on two independent computations. On the geometric side, the paper reviews Ziller's Morse-theoretic description of free loop spaces of compact symmetric spaces [MR649625], upgrades it from $\mathbb{Z}/2$ to integer coefficients for the Grassmannian, and identifies each critical submanifold of the energy functional as an explicit unitary orbit. On the algebraic side, the paper computes the $k$-colored $sl(N)$ homology of the odd-strand torus knots $T(2,2m+1)$ — new beyond the trefoil case [MR4903257] — using minimal Rickard complexes constructed in [wang2025minimalrickardcomplexesbraids]. The two computations produce summands indexed by partitions with at most $\min(k,N-k)$ parts, with matching grading shifts.

The motivating observation: $T(2,\infty)$ and $sl(N)$0

The simplest instance requires no new computation. After a downward grading shift by $sl(N)$1, the Khovanov homology of $sl(N)$2 stabilizes as $sl(N)$3 to the stable Khovanov homology of $sl(N)$4, a chain-level colimit that also appears as the unknot invariant in the Cooper–Krushkal categorification of the colored Jones polynomial [MR2901969] and as the subject of the Gorsky–Oblomkov–Rasmussen conjecture [MR3171092]. Up to grading shifts, this stable group contains only two types of direct summands: one copy of the cohomology of $sl(N)$5, and infinitely many copies of the cohomology of $sl(N)$6 shifted by $sl(N)$7 for $sl(N)$8. Identifying these summands with $sl(N)$9 and $T(2,\infty)$0 goes back to Kronheimer–Mrowka [MR2860345] and Jacobsson–Rubinsztein (0806.2902).

The key observation is that $T(2,\infty)$1 is precisely the space of closed geodesics on the round $T(2,\infty)$2-sphere: the copy of $T(2,\infty)$3 consists of constant loops, and the $T(2,\infty)$4th copy of $T(2,\infty)$5 consists of geodesics traversing a great equator $T(2,\infty)$6 times. Since the energy functional on $T(2,\infty)$7 is a perfect Morse–Bott function,

$T(2,\infty)$8

which matches the stable Khovanov homology after collapsing the bigrading appropriately. The bigraded refinement uses the Gugenheim–May theorem [MR0394720]: for a homogeneous space $T(2,\infty)$9, ordinary cohomology is recovered from equivariant cohomology not by naive tensor product but by the torsion product $\Gr(k,N)$0, whose bigrading reproduces exactly the Khovanov bigrading convention.

Loop homology of the Grassmannian

Let $\Gr(k,N)$1 with its standard symmetric-space metric, realized as $\Gr(k,N)$2. The paper gives an explicit root-type decomposition of $\Gr(k,N)$3 relative to a maximal abelian subspace $\Gr(k,N)$4, identifies the lattice of closed geodesics in the flat totally geodesic torus, and obtains a bijection between partitions $\Gr(k,N)$5 with at most $\Gr(k,N)$6 parts (assuming $\Gr(k,N)$7; otherwise use orthogonal complements) and connected components $\Gr(k,N)$8 of the space of closed geodesics. Each $\Gr(k,N)$9 is the homogeneous space $\mathbb{Z}/2$0, where $\mathbb{Z}/2$1 depends only on the multiplicities $\mathbb{Z}/2$2 of the parts of $\mathbb{Z}/2$3:

$\mathbb{Z}/2$4

The stabilizer computation proceeds via the adjoint action on $\mathbb{Z}/2$5; the proof that $\mathbb{Z}/2$6 is clean and self-contained. Ziller's theorem then yields, after upgrading coefficients to $\mathbb{Z}/2$7 (the upgrade requires only orientability of the Bott–Samelson "completing manifolds" $\mathbb{Z}/2$8, which follows because $\mathbb{Z}/2$9 is connected):

$k$0

where $k$1 with grading collapsed by $k$2 and negated. The shift exponent follows from the explicit Morse index formula $k$3.

Colored link homology of $k$4

The second half computes the $k$5-colored $k$6 complexes of $k$7 and $k$8 in terms of chain complexes $k$9 of $sl(N)$0-modules. The central technical object is a complex $sl(N)$1 modeled on the cube $sl(N)$2, built from the polynomial ring $sl(N)$3 with differential components given by composites of divided differences $sl(N)$4, transpositions $sl(N)$5, and the element $sl(N)$6. The paper constructs a genuine action of the nil-Hecke algebra $sl(N)$7 on $sl(N)$8 by chain maps — a nontrivial point, since the naive per-vertex action does not commute with the differential. The generators are modified endomorphisms $sl(N)$9 and $T(2,2m+1)$0, where $T(2,2m+1)$1 has components along non-edge diagonals of the cube. Verification of the nil-Hecke relations occupies a lengthy appendix of case-by-case computations using braid and far-commutativity relations.

From this action, the paper shows that $T(2,2m+1)$2 is free of graded rank $T(2,2m+1)$3 over $T(2,2m+1)$4 and admits subcomplexes $T(2,2m+1)$5 (invariants under Young subgroups) that are direct summands via standard nil-Hecke idempotents. Each $T(2,2m+1)$6 is shown to be a free resolution of the quotient imposing $T(2,2m+1)$7, hence $T(2,2m+1)$8 is an $T(2,2m+1)$9-free resolution of $\min(k,N-k)$0.

On the link side, the twist closure of the minimal Rickard complex $\min(k,N-k)$1 computes $\min(k,N-k)$2. Via deformation retracts onto single-grading webs $\min(k,N-k)$3 and the homological perturbation lemma, the complex splits as a direct sum of subcomplexes $\min(k,N-k)$4 indexed by partitions with $\min(k,N-k)$5, and each $\min(k,N-k)$6 is identified with $\min(k,N-k)$7. The main theorem follows:

$\min(k,N-k)$8

with the finite-level version restricting to $\min(k,N-k)$9. This coincides term-by-term with the loop-homology decomposition, proving the identification of stable colored $T(2,\infty)$0 homology of $T(2,\infty)$1 with $T(2,\infty)$2 up to the stated shifts. Note the grading caveat recorded in the paper's Remark: after collapsing and negating the bigrading, the graded groups agree only modulo a cyclic reduction to $T(2,\infty)$3 gradings, since summands differ by shifts $T(2,\infty)$4; without this reduction the groups are no longer finitely generated in each degree.

Limitations and open questions

The author states plainly that the coincidence between link homology and loop homology is unexplained conceptually: the proof is entirely computational, and no mechanism relating foam categories to Morse theory on loop spaces is identified. The appearance of the number $T(2,\infty)$5 in the grading discrepancy suggests a possible role for Floer theory — cyclic $T(2,\infty)$6 gradings arise in $T(2,\infty)$7 instanton Floer homology [MR2860345] and in Lagrangian Floer homology in monotone symplectic manifolds of minimal Chern number $T(2,\infty)$8 [MR987770, MR1223659] — but the paper explicitly refrains from further speculation for lack of evidence. Two additional boundaries of the result deserve note: the identification concerns only the symmetric projector (odd strands $T(2,\infty)$9), and is distinct from the Rozansky/Przytycki connection between $sl(N)$00 and Hochschild homology of $sl(N)$01, which involves the antisymmetric projector; and stable Khovanov homology of $sl(N)$02 has been computed explicitly only for $sl(N)$03 prior to this work, so the general-$sl(N)$04 analogue of the loop-space picture remains open.

Conclusion

The paper proves that stable $sl(N)$05-colored $sl(N)$06 homology of $sl(N)$07 equals the integral homology of $sl(N)$08, with matching partition-indexed decompositions and grading shifts, and supplies the first computation of colored $sl(N)$09 homology of $sl(N)$10 beyond the trefoil. The nil-Hecke action on $sl(N)$11 by chain maps is the load-bearing algebraic input, and the coefficient upgrade of Ziller's theorem is the geometric one. Whether the isomorphism reflects a deeper structure — perhaps mediated by a $sl(N)$12-graded Floer-type theory — remains the central question left open.