Exponential growth of colored HOMFLY-PT homology

Abstract

We define reduced colored sl N link homologies and use deformation spectral sequences to characterize their dependence on color and rank. We then define reduced colored HOMFLY-PT homologies and prove that they arise as large N limits of sl N homologies. Together, these results allow proofs of many aspects of the physically conjectured structure of the family of type A link homologies. In particular, we verify a conjecture of Gorsky, Gukov and Stošić about the growth of colored HOMFLY-PT homologies. arXiv:1602.02769v1 [math.GT] 8 Feb 2016 Theorem 2. . Let K be a labelled knot. For sufficiently large N , there are isomorphisms i+2N j=I h−j=Jbetween a grading-collapsed version of reduced colored HOMFLY-PT homology and reduced colored sl N homology. See Theorem 3.46.Theorem 2 relies on the fact that the reduced colored HOMFLY-PT homology is finite-dimensional for knots, see Proposition 3.45. On the contrary, the original unreduced colored HOMFLY-PT homology of Mackaay-Stošić-Vaz [36] and Webster-Williamson [53] is always infinite-dimensional. Nevertheless, we find an analogue of Theorem 1, which relates it to unreduced colored sl N homologies and -more generally -their Σ-deformed versions, which have been defined by Wu in [58] and further studied by Rose and the author in [49].Theorem 3. Let L be a labelled link. There is a spectral sequencefrom unreduced colored HOMFLY-PT homology to unreduced, Σ-deformed colored sl N homology. See Theorem 3.22.1.2. The physical structure. The existence of reduced colored sl N and HOMFLY-PT homologies and the theorems describing their relationships, although new, will not come as a surprise to the expert. In fact, they are at the very centre of current interest in the study of link homologies from the prespective of theoretical physics. Since Gukov-Schwarz-Vafa's physical interpretation of Khovanov-Rozansky homology as a space of BPS states [22], there has been significant cross-fertilization between the mathematical and physical sides of this field. A particularly interesting outcome is a package of conjectures about the structure of the family of colored sl N and HOMFLY-PT homologies, which has been developed over a decade in a series of papers by Dunfield-Gukov-Rasmussen [12], Gukov-Walcher [24], Gukov-Stošić [23], Gorsky-Gukov-Stošić [18] and most recently Gukov-Nawata-Saberi-Stošić-Su lkowski [20]. Thanks to recent advances in understanding the higher representation theory underlying link homologies (in particular categorical skew Howe duality), which led to the convergence of several different approaches to link homologies [7,42,49,37], we are now in a position to prove a significant part of the physically conjectured structure. In the following theorems, K k denotes a knot labelled with the k th exterior power of the vector representation.Theorem 4. There is a spectral sequencewhich preserves the homological grading. See Corollary 3.47.This implies that the colored HOMFLY-PT homologies of a knot grow at least exponentially in color, as predicted in [18]. A related result has ...

Knots-quivers correspondence

“…Furthermore, HOMFLY-PT homologies of a large class of knots satisfy the refined exponential growth, which implies the following relation for their colored superpolynomials P S r (a, q = 1, t) = (P (a, q = 1, t)) r , (2.13) see also [35]. Properties of colored differentials, together with the assumption of the exponential growth, enable to determine an explicit form of the colored superpolynomial P S r (a, q, t) for various knots [38,45,48].…”

“…It has been defined rigorously by mathematicians only recently [34], yet only for the unreduced version, and it is still not suitable for explicit computations (there also exist some constructions in the case of antisymmetric representations Λ r , both reduced and unreduced versions, see e.g. [35], which are conjecturally isomorphic to the homologies corresponding to the symmetric representations). Nonetheless, the conjectural Poincaré polynomial of (reduced) colored HOMFLY-PT homology, referred to as the superpolynomial…”

Upsilon-like concordance invariants from 𝔰𝔩n knot cohomology

“…A proof of this can be found in the proof of Theorem 2.15 of [Wed16], which considers colored perturbed sl n cohomology of a (1, 1)-tangle (which for us is the diagram D cut open at the basepoint). Specializing to the 1-colored case and working with a general degree n potential ∂w, Theorem 2.15 identifies the cohomology of the λ-eigenspace of the complex C ∂w (D) with the sl m cohomology of D where m is the multiplicity of λ as a root of ∂w.…”

Functoriality of colored link homologies

“…For example, Hedden–Ni have shown that they enable Khovanov homology to detect unlinks (it was later proven that the module structure is determined by the bigraded vector space structure in the case of the unlink ), see also Wu for a related result for the triply‐graded homology. Module structures are also important for the comparison with Floer‐theoretic link invariants, see for example, Baldwin–Levine–Sarkar , and the construction of reduced , colored Khovanov–Rozansky homologies .…”