Gukov–Manolescu Series: Knot and 3-Manifold Invariants

• The Gukov–Manolescu series is a two-variable formal q-series invariant of knot complements that re-sums the Melvin–Morton–Rozansky expansion and connects colored Jones polynomials to 3-manifold invariants.
• It is constructed via inverted state sums and Habiro series, yielding explicit coefficient formulas that relate to the Alexander polynomial and capture complex analytic structures.
• Recent advancements extend the framework to higher rank, symmetric representations, and supergroup analogues, providing surgery formulas and insights into topological invariants of plumbed 3-manifolds.

The Gukov–Manolescu series is a two-variable formal series attached to a knot complement, usually written $F_K(x,q)$, proposed as a re-summation of the Melvin–Morton–Rozansky expansion of colored Jones polynomials and as the knot-complement analogue of the 3-manifold invariant $\widehat Z$. In the $\mathfrak{sl}_2$ setting it is a series in a boundary variable and the $q$-parameter, while later work extended the construction to links, to higher rank, to root-lattice twists, and to supergroup analogues. In adjacent literature, the same circle of ideas also includes homological blocks $\widehat Z_a(q)$ for plumbed 3-manifolds, so the phrase “Gukov–Manolescu series” is sometimes used in a broader sense that includes both knot-complement and plumbed-manifold $q$-series (Park, 2021, Moore et al., 6 Jun 2025, Park, 2019).

1. Definition, normalizations, and relation to large-color expansions

A standard normalization writes the knot series as

$$F_K(x,q)=(x^{1/2}-x^{-1/2})\,Z^{\mathrm{inv}}(\beta),$$

where $Z^{\mathrm{inv}}(\beta)$ is an inverted state sum associated to a braid representative $\beta$ of the knot or link (Park, 2021). In another normalization, used for the positive part of the series, one writes

$x^{-1/2}F_K(x,q)=\sum_{n\ge 0} f_{K;n}(q)\,x^n,$

with coefficients $\widehat Z$0; for knots that are closures of homogeneous braids, these coefficients are Laurent polynomials (Svoboda, 26 Sep 2025). For “nice” knots, the series is also written in the form

$\widehat Z$1

where $\widehat Z$2 and $\widehat Z$3 (Orland et al., 29 Dec 2025).

The defining motivation is the large-color regime of the colored Jones polynomial. The standard variables are

$\widehat Z$4

and the Gukov–Manolescu series is expected to reorganize the Melvin–Morton–Rozansky expansion into an integral two-variable $\widehat Z$5-series (Orland et al., 13 Jul 2026). In the homogeneous-braid setting, the inverted state sum is proved to agree term-by-term in $\widehat Z$6 with the Melvin–Morton–Rozansky expansion, thereby proving the Gukov–Manolescu conjecture for homogeneous braid links (Park, 2021). A basic specialization is the $\widehat Z$7 limit: $\widehat Z$8 and

$\widehat Z$9

so the Alexander polynomial controls the leading classical behavior (Orland et al., 13 Jul 2026).

This relation to the Alexander polynomial is also visible in the semiclassical limit of higher-rank analogues. For $\mathfrak{sl}_2$0, one has

$\mathfrak{sl}_2$1

which is the direct $\mathfrak{sl}_2$2 analogue of the $\mathfrak{sl}_2$3 structure (Gruen et al., 21 Aug 2025).

2. Inverted state sums, inverted Habiro series, and regularization

For homogeneous braid links, and more generally for braids admitting an appropriate sign assignment on segments, the series can be constructed by an inverted large-color $\mathfrak{sl}_2$4-matrix state sum. In the algorithmic formulation, given a braid $\mathfrak{sl}_2$5 and an acceptable inversion datum $\mathfrak{sl}_2$6, one defines

$\mathfrak{sl}_2$7

where $\mathfrak{sl}_2$8 is the constrained state space and $\mathfrak{sl}_2$9 is the product of local $q$0-matrix weights and reduced-trace factors (Orland et al., 13 Jul 2026). In Park’s general theorem, the required conditions are that the number of incoming and outgoing $q$1-signed segments agree at each crossing and that the partial quantum trace computed after inverting the $q$2-signed segments converges absolutely (Park, 2021).

A second description uses the inverted Habiro series. For a knot $q$3,

$q$4

and the coefficients $q$5 are equivalent to the GM coefficients $q$6 through explicit $q$7-binomial transforms (Svoboda, 26 Sep 2025). Concretely,

$q$8

This makes the inverted Habiro coefficients and the GM coefficients mutually determining (Svoboda, 26 Sep 2025).

The analytic point is that the formal $q$9-series $\widehat Z_a(q)$0 can be badly behaved as a $\widehat Z_a(q)$1-Laurent series. For generic $\widehat Z_a(q)$2, the radius of convergence in $\widehat Z_a(q)$3 is typically zero; for the figure-eight knot $\widehat Z_a(q)$4,

$\widehat Z_a(q)$5

The inverted Habiro series compensates for these increasingly negative $\widehat Z_a(q)$6-powers. Under the lower bound condition

$\widehat Z_a(q)$7

the series defines an element of $\widehat Z_a(q)$8 and admits a closed expression in terms of Jacobi theta functions: $\widehat Z_a(q)$9 This identity shows that the truncated theta functions cancel the increasingly negative $q$0-powers occurring in the GM coefficients (Svoboda, 26 Sep 2025).

3. Surgery, residues, and analytic structure

A central structural feature is the surgery transform from knot complements to closed 3-manifolds. In the Gukov–Manolescu framework, for integer surgery one has

$q$1

and, in terms of the coefficients $q$2,

$q$3

The same paper derives an explicit residue formula when the relevant double series converges: $q$4 where $q$5 (Svoboda, 26 Sep 2025).

The residue calculus is built directly into the inverted Habiro series. For $q$6, the meromorphic 1-form $q$7 has simple poles at $q$8, and the residues of the full series are

$q$9

These residues reconstruct both the pole expansion of $$F_K(x,q)=(x^{1/2}-x^{-1/2})\,Z^{\mathrm{inv}}(\beta),$$0 and the GM coefficients, since for $$F_K(x,q)=(x^{1/2}-x^{-1/2})\,Z^{\mathrm{inv}}(\beta),$$1,

$$F_K(x,q)=(x^{1/2}-x^{-1/2})\,Z^{\mathrm{inv}}(\beta),$$2

The residue theorem then yields

$$F_K(x,q)=(x^{1/2}-x^{-1/2})\,Z^{\mathrm{inv}}(\beta),$$3

under the lower bound condition (Svoboda, 26 Sep 2025).

Regularized surgery formulas already appeared in the inverted-state-sum approach. In particular, for $$F_K(x,q)=(x^{1/2}-x^{-1/2})\,Z^{\mathrm{inv}}(\beta),$$4 surgery,

$$F_K(x,q)=(x^{1/2}-x^{-1/2})\,Z^{\mathrm{inv}}(\beta),$$5

and for $$F_K(x,q)=(x^{1/2}-x^{-1/2})\,Z^{\mathrm{inv}}(\beta),$$6 surgery,

$$F_K(x,q)=(x^{1/2}-x^{-1/2})\,Z^{\mathrm{inv}}(\beta),$$7

These formulas introduce false-theta-type regulators and lead to expressions of $$F_K(x,q)=(x^{1/2}-x^{-1/2})\,Z^{\mathrm{inv}}(\beta),$$8 for some plumbed 3-manifolds in terms of indefinite theta functions (Park, 2021).

A notable asymptotic phenomenon occurs at roots of unity. For the figure-eight knot $$F_K(x,q)=(x^{1/2}-x^{-1/2})\,Z^{\mathrm{inv}}(\beta),$$9, the Kashaev invariant satisfies

$Z^{\mathrm{inv}}(\beta)$0

while the GM coefficients satisfy

$Z^{\mathrm{inv}}(\beta)$1

The quotient $Z^{\mathrm{inv}}(\beta)$2 has integral Taylor coefficients, and the paper records numerical evidence for periodic phenomena such as periodicity of $Z^{\mathrm{inv}}(\beta)$3 for $Z^{\mathrm{inv}}(\beta)$4 (Svoboda, 26 Sep 2025).

4. Examples, coefficient formulas, and computation

The standard examples already exhibit the principal behaviors of the theory. For the figure-eight knot $Z^{\mathrm{inv}}(\beta)$5,

$Z^{\mathrm{inv}}(\beta)$6

and

$Z^{\mathrm{inv}}(\beta)$7

The corresponding residues have both single-sum and double-sum descriptions, and the initial terms of $Z^{\mathrm{inv}}(\beta)$8 and $Z^{\mathrm{inv}}(\beta)$9 are recorded explicitly (Svoboda, 26 Sep 2025). For the left-handed trefoil $\beta$0,

$\beta$1

while for the right-handed trefoil $\beta$2,

$\beta$3

In the latter case the inverted Habiro series admits a Ramanujan-type expression

$\beta$4

and the residue theorem yields a Hecke–Rogers identity (Svoboda, 26 Sep 2025).

In the original inverted-state-sum computations, the figure-eight knot, knots $\beta$5 and $\beta$6, the Whitehead link, the Borromean rings, and the two-component link $\beta$7 were all given explicitly. For example,

$\beta$8

and its $\beta$9-expansion begins with the coefficients displayed in the paper (Park, 2021).

Systematic computation is now available from braid data. The package fkcompute implements Park’s inverted state sum through a three-phase pipeline. First, a search is performed for a suitable braid presentation and for an additional inversion datum on the braid. Then, the state space of the inverted sum is encoded as a polytope, bounded by the associated linear constraint system. Finally, the invariant is constructed by multiplication of $x^{-1/2}F_K(x,q)=\sum_{n\ge 0} f_{K;n}(q)\,x^n,$0-matrices associated to the states. Benchmarks show that prime knots up to 12 crossings, and prime links up to 10 crossings and of at most 3 components, are comfortably within reach, and the package has been used to compile the first public database of the Gukov–Manolescu invariant (Orland et al., 13 Jul 2026).

The same work also makes the state-sum mechanics explicit. The braid is oriented bottom to top, states assign integers to segments, and the constrained state space is encoded as integer points in a bounded polyhedron

$x^{-1/2}F_K(x,q)=\sum_{n\ge 0} f_{K;n}(q)\,x^n,$1

The reduced-trace normalization fixes a ground state on the open strand and inserts factors

$x^{-1/2}F_K(x,q)=\sum_{n\ge 0} f_{K;n}(q)\,x^n,$2

for the closed strands. Enumeration of integer points in $x^{-1/2}F_K(x,q)=\sum_{n\ge 0} f_{K;n}(q)\,x^n,$3, together with evaluation of local $x^{-1/2}F_K(x,q)=\sum_{n\ge 0} f_{K;n}(q)\,x^n,$4-matrix factors and $x^{-1/2}F_K(x,q)=\sum_{n\ge 0} f_{K;n}(q)\,x^n,$5-Pochhammer expansions, yields truncated expansions of $x^{-1/2}F_K(x,q)=\sum_{n\ge 0} f_{K;n}(q)\,x^n,$6 or $x^{-1/2}F_K(x,q)=\sum_{n\ge 0} f_{K;n}(q)\,x^n,$7 in sparse polynomial form (Orland et al., 13 Jul 2026).

5. Higher-rank, symmetric-color, and supergroup extensions

A higher-rank extension replaces the $x^{-1/2}F_K(x,q)=\sum_{n\ge 0} f_{K;n}(q)\,x^n,$8 data by a connected, simply-connected semisimple Lie group $x^{-1/2}F_K(x,q)=\sum_{n\ge 0} f_{K;n}(q)\,x^n,$9 with root system $\widehat Z$00. In the plumbed-manifold setting the labels become

$\widehat Z$01

which generalizes Spin$\widehat Z$02 structures and carries a Weyl-group action (Park, 2019). For weakly negative-definite plumbed 3-manifolds, the higher-rank homological block is defined by a principal-value contour integral

$\widehat Z$03

and is invariant under Neumann moves (Park, 2019). In the same paper, $\widehat Z$04 is defined for torus-knot complements, and a higher-rank surgery formula relates $\widehat Z$05 to $\widehat Z$06 of Dehn surgeries (Park, 2019).

Specialization to symmetric representations leads to an $\widehat Z$07 large-color $\widehat Z$08-matrix formalism. For positive braid knots, the symmetrically large coloured $\widehat Z$09-matrix defines $\widehat Z$10, and the resulting series is annihilated by the $\widehat Z$11 specialization of the quantum $\widehat Z$12-polynomial. The same work formulates a three-variable $\widehat Z$13-deformed conjecture $\widehat Z$14 interpolating all $\widehat Z$15 and satisfying Weyl symmetry

$\widehat Z$16

For $\widehat Z$17 torus knots the paper gives explicit quiver descriptions and block-structured quiver matrices (Gruen, 2022, Kucharski, 2020).

The first non-symmetric full-rank construction appears for $\widehat Z$18. For a positive braid knot $\widehat Z$19,

$\widehat Z$20

with $\widehat Z$21, and

$\widehat Z$22

The construction uses two symmetric Verma modules $\widehat Z$23 and $\widehat Z$24, computes the large-color $\widehat Z$25-matrix on $\widehat Z$26, and recovers the reduced colored $\widehat Z$27 Reshetikhin–Turaev invariant after specialization $\widehat Z$28, $\widehat Z$29 (Gruen et al., 21 Aug 2025).

A different extension replaces semisimple Lie algebras by the Lie superalgebra $\widehat Z$30. For plumbed knot complements, the supergroup series is a three-variable invariant

$\widehat Z$31

built from chamber-dependent contour integrals and lattice theta functions. Its general form is

$\widehat Z$32

and it satisfies the Weyl symmetry

$\widehat Z$33

The corresponding Dehn surgery formula sends $\widehat Z$34 to the Ferrari–Putrov closed-manifold invariant $\widehat Z$35 by chamber-dependent Laplace transforms, and the paper computes explicit examples for $\widehat Z$36, $\widehat Z$37, and $\widehat Z$38 (Chae, 14 Aug 2025).

6. Plumbed 3-manifolds, topological interpretations, and conjectural directions

In the plumbed-manifold setting, the Gukov–Manolescu programme interacts with homological blocks, root-lattice twists, and lattice-homological models. For reduced refinable plumbing trees and a root lattice $\widehat Z$39, one defines

$\widehat Z$40

and for the Kostant collection $\widehat Z$41 and the full set $\widehat Z$42 of Weyl assignments,

$\widehat Z$43

This series is the unique $\widehat Z$44-invariant average among the Neumann-invariant series of this form (Moore et al., 2024). In a further extension, refined series $\widehat Z$45 and $\widehat Z$46 for plumbed 3-manifolds and knot complements are defined for arbitrary root lattices and non-definite plumbings, satisfy gluing and splitting formulas, and reduce to the GPPV/Gukov–Manolescu series in the negative-definite case at $\widehat Z$47 (Moore et al., 6 Jun 2025).

The relation to lattice homology is made precise by a weighted bigraded root. For a negative definite marked plumbing graph $\widehat Z$48, the weighted bigraded root simultaneously encodes the degree-zero part of knot lattice homology and the Gukov–Manolescu homological block of the plumbed knot complement. In the stabilized $\widehat Z$49-direction, its weights recover

$\widehat Z$50

and a surgery formula at the level of weighted roots reproduces the Gukov–Manolescu integer surgery transform in the stabilized limit (Akhmechet et al., 2024).

Indefinite plumbings require additional regularization. For plumbing indefinite $\widehat Z$51-graphs, the appropriate candidates are indefinite false theta functions

$\widehat Z$52

built from a two-dimensional indefinite quadratic form and a positive-cone regulator. They satisfy the radial-limit identity

$\widehat Z$53

and for the Poincaré homology sphere the resulting indefinite false theta function coincides with the original homological block (Murakami, 2022).

Recent work also ties the knot-complement series more directly to topological features of knots. For a nice knot,

$\widehat Z$54

so the leading coefficient is a monomial in $\widehat Z$55. For homogeneous braid knots, and for fibered knots with at most 12 crossings, one has

$\widehat Z$56

where $\widehat Z$57 is the Hopf invariant. The same paper proves that all fibered knots up to 12 crossings are nice, proposes that a knot is nice if and only if it is fibered, and formulates a slope conjecture

$\widehat Z$58

relating $\widehat Z$59-slopes to boundary slopes (Orland et al., 29 Dec 2025).

A persistent misconception is that the Gukov–Manolescu series is already a uniformly convergent analytic function in both variables for arbitrary knots. The literature instead distinguishes sharply between formal $\widehat Z$60-series, meromorphic regularizations such as the inverted Habiro series, state-sum definitions available for special braid classes, and broader conjectural extensions. A plausible implication is that the mature part of the theory lies not in a single universal closed formula, but in the compatibility among re-summation, state-sum, residue, surgery, and root-lattice constructions across overlapping domains of definition (Svoboda, 26 Sep 2025, Orland et al., 13 Jul 2026).