Homological Blocks in Quantum Topology

• Homological blocks are q-series invariants attached to 3-manifolds, defined as families of series whose radial limits recover WRT invariants and other quantum-topological measures.
• In the Seifert-fibered setting, these blocks are computed using iterative Atiyah–Bott fixed point formulas and Lie algebra techniques to yield explicit q-series representations.
• For knot complements, the homological block appears as an inverted Habiro series and half-index, representing the contribution of the abelian flat connection with links to colored Jones polynomials.

Homological block denotes a class of $q$-series attached to $3$-manifolds that, in the current literature represented here, is studied most explicitly for Seifert fibered homology $3$-spheres and for knot complements in $S^3$. In the Gukov–Pei–Putrov–Vafa framework cited by later work, these series are expected to satisfy radial limits reproducing the Witten–Reshetikhin–Turaev (WRT) invariants; in the Seifert setting with simple Lie algebra coefficients, that relation is proved, while for knot complements the block appears as a $q,x$-series realized as a half-index and interpreted as the contribution associated to the abelian flat connection (Sugimoto, 2022, Murakami et al., 2023, Chung, 5 Mar 2026).

1. Scope of the term

The literature represented here uses “homological block” in several closely related but not identical settings. For plumbed $3$-manifolds, and in particular Seifert fibered homology $3$-spheres, homological blocks are $q$-series expected to have radial limits reproducing WRT invariants, often resembling false/mock theta functions or more general quantum modular objects, and they can be interpreted as characters of certain modules of logarithmic VOAs (Sugimoto, 2022). For knot complements, the basic object is the homological block

$F_K(x,q),$

a $q,x$-series which, for $3$0, has a balanced expansion, a Laurent series in $3$1 with coefficients in $3$2, invariant under the Weyl action $3$3, together with positive and negative expansions given by the positive and negative $3$4-power parts of the balanced expansion (Chung, 5 Mar 2026).

Setting	Basic object	Role stated in the literature
Seifert fibered homology $3$5-sphere	homological block, or $3$6 with simple Lie algebra data	radial limits reproduce or are identified with WRT invariants
Knot complement $3$7	$3$8	half-index of a $3$9 theory and contribution associated to the abelian flat connection

This distribution of usages suggests that “homological block” is not a single universal formula but a family of $3$0-series constructions adapted to different $3$1-manifold problems. A plausible implication is that the unifying content is the role of these series as boundary or holomorphic objects whose radial or contour limits recover quantum-topological invariants.

2. Seifert fibered homology $3$2-spheres

One principal setting is the Seifert fibered homology $3$3-sphere. In one formulation, the manifolds considered are

$3$4

with $3$5, $3$6, pairwise coprime $3$7, and

$3$8

These are exactly the Seifert fibered homology $3$9-spheres for which generalized homological blocks are defined in the simple-Lie-algebra treatment (Murakami et al., 2023).

A complementary representation-theoretic description treats Seifert fibered homology $S^3$0-spheres with $S^3$1 fibers. There, the homological block is computed by repeatedly applying the Atiyah–Bott fixed point formula to a sequence of equivariant bundle constructions modeled on Feigin–Tipunin-type geometric realizations of logarithmic VOA modules. The calculation starts with a lattice VOA module or theta-function-like object, applies a geometric functor $S^3$2 repeatedly, and uses the Atiyah–Bott fixed point formula at each stage to compute characters, with the final output expressed as an explicit $S^3$3-series or lattice-sum formula (Sugimoto, 2022).

This Seifert-fibered setting is also where several structural expectations become concrete. The paper on Atiyah–Bott methods explicitly relates $S^3$4-fibered Seifert manifolds to $S^3$5-log VOAs and $S^3$6-fibered Seifert manifolds to $S^3$7-log VOAs, and suggests that higher-fiber cases should correspond to iterated versions of these constructions (Sugimoto, 2022). That perspective places homological blocks at the intersection of quantum topology, geometric representation theory, and logarithmic VOA character theory.

3. Simple Lie algebra generalization and radial limits

For a complex simple Lie algebra $S^3$8 with positive roots $S^3$9, the Seifert-fibered construction defines a generalized homological block by

$q,x$0

where $q,x$1 is the Weyl vector, $q,x$2 is the norm induced by the standard inner product normalized so that the longest roots have length $q,x$3, and $q,x$4, $q,x$5, and

$q,x$6

are determined explicitly in the construction (Murakami et al., 2023).

The main theorem states that for every positive integer $q,x$7, the radial limit $q,x$8 is expressed by a finite lattice sum with a Gaussian factor and a product of explicit one-variable functions $q,x$9. The central comparison is with a formula of Mariño for the WRT invariant $3$0 when $3$1 is simply-laced, and the consequence is that for simply-laced $3$2, the WRT invariant is a scalar multiple of the radial limit of the generalized homological block (Murakami et al., 2023).

Several specializations anchor the construction. For $3$3, the block reduces to the original homological blocks of Gukov–Pei–Putrov–Vafa. For $3$4, it agrees with the $3$5 generalization previously defined by Chung (Murakami et al., 2023). Thus the simple-Lie-algebra generalization is simultaneously a genuine extension and a compatibility result with earlier cases.

4. Computational mechanisms: Atiyah–Bott iteration and asymptotic analysis

The representation-theoretic computation of Seifert homological blocks uses a simply-laced simple Lie algebra $3$6, a triangular decomposition

$3$7

a Borel subgroup $3$8, root lattice $3$9, weight lattice $3$0, dominant weights $3$1, Weyl group $3$2, Weyl vector $3$3, and a lattice VOA built from the rescaled root lattice $3$4, with $3$5. A central technical input is Felder exactness: a family of $3$6-modules with screening operators $3$7 is called Felder exact if the kernels are $3$8-submodules and if short exact sequences

$3$9

exist in the stated form (Sugimoto, 2022).

The geometric functor is $q$0. The evaluation map

$q$1

is injective, $q$2 can be identified with a maximal $q$3-submodule of $q$4, and higher cohomology frequently vanishes: $q$5 That vanishing is what allows the Atiyah–Bott fixed point formula to compute $q$6 from the full character of the ambient $q$7-module, and the novelty is that the one-step procedure is repeated $q$8 times (Sugimoto, 2022).

The analytic route developed in the simple-Lie-algebra paper is different but complementary. It proves a new asymptotic expansion for a class of partial theta-type series, formulated in Poincaré’s sense, and combines this with a vanishing and holomorphy result for asymptotic coefficients. The key point is that the asymptotic expansion simplifies because all the “bad” coefficients cancel or vanish, leaving precisely the finite lattice sum needed for the WRT formula (Murakami et al., 2023). Taken together, these two approaches exhibit homological blocks both as iterated character computations and as analytically controlled partial-theta-type series.

5. Knot complements, inverted Habiro series, and the $q$9–$F_K(x,q),$0 correspondence

For a knot complement

$F_K(x,q),$1

with gauge group $F_K(x,q),$2, the homological block is the $F_K(x,q),$3-series $F_K(x,q),$4. In the $F_K(x,q),$5–$F_K(x,q),$6 correspondence, the analytically continued Chern–Simons partition function on $F_K(x,q),$7 is related to the half-index of a $F_K(x,q),$8 theory $F_K(x,q),$9 on $q,x$0 with boundary conditions preserving $q,x$1 supersymmetry. The homological block is interpreted as the contribution associated to the abelian flat connection, but it also encodes non-abelian information through resurgence (Chung, 5 Mar 2026).

The paper realizes the homological block as an inverted Habiro series. Starting from the Habiro cyclotomic expansion of the colored Jones polynomial, it proposes the normalized homological block in the form

$q,x$2

The central structural point is that the homological block is the inverted Habiro series, whereas the colored Jones polynomial is the Habiro expansion; by changing which poles are selected in the half-index integral, one can move between these two (Chung, 5 Mar 2026).

Worked examples include the figure-eight knot and the left- and right-handed trefoils. For the figure-eight knot,

$q,x$3

and the classical $q,x$4-polynomial contains the explicit abelian branch factor $q,x$5. In the half-index integral, choosing poles

$q,x$6

coming from $q,x$7 yields the homological block, whereas choosing

$q,x$8

from $q,x$9, and then setting $3$00, yields the colored Jones polynomial (Chung, 5 Mar 2026). This identifies contour choice with the abelian/non-abelian split in a precise operational way.

6. Conceptual status, limitations, and terminological boundaries

The papers considered here do not present a single finalized universal theory of homological blocks. Rather, they present a cluster of constructions with varying theorem-strength. In the Seifert-fibered setting with simple Lie algebra coefficients, the statement that radial limits are identified with WRT invariants is proved, and the proof uses both an asymptotic formula and a vanishing result of asymptotic coefficients (Murakami et al., 2023). In the knot-complement setting, the homological block is realized as a half-index by working out examples, “which we expect to extend to general knots” (Chung, 5 Mar 2026). This suggests that the concept is already rigid enough to support exact comparison theorems in some settings, while remaining partly conjectural or example-driven in others.

A recurrent misconception is to treat every mathematical use of the word “block” as relevant. The supplied source “A generalised block decomposition theorem” concerns a class of linear representations of the product poset $3$01 that decompose into interval representations for block intervals, characterized by a homological property called middle exactness in the abstract, but the supplied content also states that there is “No PDF for (Lerch, 2024)v2,” that the author provided no source to generate a PDF, and that the supplied text does not define a “block interval” or state the decomposition theorem beyond the abstract. Accordingly, no substantive connection between that source and homological blocks in quantum topology can be extracted from the supplied material (Lerch, 2024).

Within the present corpus, the most stable conceptual through-line is the following: a homological block is a $3$02-series invariant attached to a $3$03-manifold or knot complement, typically organized so that radial limits, residue computations, or contour prescriptions recover WRT invariants, colored Jones polynomials, or the contribution of the abelian flat connection. That synthesis is explicit in the Seifert-fibered and knot-complement cases, and it is the sense in which the term is used in the recent literature surveyed here.