fkcompute: an efficient $F_K$ invariant calculator

Abstract: We introduce fkcompute, an open-source package for computing the Gukov--Manolescu invariant of links from a braid presentation. 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 data 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 R-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. As a result, fkcompute is used to compile the first public database of the Gukov--Manolescu invariant. The package is available as a Python library, a command-line tool, and a Mathematica paclet.

• The paper introduces an open-source calculator that computes the Gukov–Manolescu invariant from braid presentations through inversion-data search, polytope-based state encoding, and parallel large-color R-matrix evaluation.
• The software routinely handles prime knots through 12 crossings and prime links through 10 crossings, with near-linear 12-core scaling and strong validation against Alexander-polynomial and independent quantum-topology calculations.
• The paper’s database produces mathematical consequences, including evidence for slope behavior and a disproof of the conjectural meromorphicity of Park’s inverted Habiro series, while highlighting limitations for knots without acceptable inversion data.

Overview

This paper introduces fkcompute, an open-source software package for computing the Gukov–Manolescu invariant $F_K$ of knots and links from braid presentations. The implementation realizes Park's inverted state sum through a three-phase pipeline: a search for a suitable braid presentation together with inversion data, the encoding of the state space as a polytope defined by a linear constraint system, and the construction of the invariant by multiplication of large-color $R$-matrices over the integral points of that polytope. The package is distributed as a Python library, a command-line tool, and a Mathematica paclet, and is accompanied by the first publicly available database of Gukov–Manolescu series (2607.12155).

The computational reach reported is substantial: prime knots up to 12 crossings and prime links up to 10 crossings with at most 3 components are computed routinely. Beyond its engineering contribution, the package has already produced mathematical results — most notably a disproof of the conjectural meromorphicity of Park's inverted Habiro series.

Mathematical background

The $F_K$ invariant originates from the $\widehat{Z}$ program in quantum topology. The $\widehat{Z}$ invariants are $q$-series quantum invariants of closed 3-manifolds motivated by 6D supersymmetric field theory, expected to recover Witten–Reshetikhin–Turaev invariants via radial limits at roots of unity and to admit a categorification analogous to Khovanov homology. Gukov and Manolescu proposed $F_L$, a series in $\ell+1$ variables ($\ell$ being the number of link components), as the analogue for link complements.

Two structural relations constrain $F_K$ and serve both as motivation and as validation targets. First, closed 3-manifold invariants should be recoverable from link-complement series by Dehn surgery: $R$0 is expressed as a Laplace transform applied to $R$1. A fully generic expression for this transform has not been found, which the authors identify as one of the principal motivations for tabulating explicit values of $R$2. Second, Park proved that for his class of nice knots, $R$3 reorganizes the Melvin–Morton–Rozansky (MMR) expansion of the colored Jones polynomial into an integral two-variable $R$4-series; in particular, at $R$5 one recovers the inverse Alexander polynomial.

The inverted state sum

Computationally, fkcompute follows Park's large-color $R$6-matrix construction, formally parallel to Reshetikhin–Turaev evaluation of the colored Jones polynomial. A braid $R$7 on $R$8 strands is decorated by integer state labels on its segments; each crossing contributes an extended $R$9-matrix entry depending on the four adjacent labels and the topological variables assigned to the two strands passing through it, built from $F_K$0-binomial coefficients and (possibly inverted) $F_K$1-Pochhammer factors. State labels satisfy a conservation law at each crossing ($F_K$2).

The key combinatorial input is an inversion datum: a sign assignment on segments specifying whether each state variable ranges over non-negative or negative integers, which determines where the implicit geometric series expansions are inverted. An inversion datum is acceptable when every fixed-$F_K$3-degree coefficient of the resulting sum is a finite Laurent polynomial in $F_K$4. Knots admitting such data are called nice. Homogeneous braids carry a canonical inversion datum via Park's rule; for non-homogeneous braids the datum must be searched for. The authors note plainly that the geometric meaning of the inversion datum is not yet fully understood — its role here is to make the formal sum computable by finite means.

The final invariant is obtained from the reduced quantum trace: the ground state is fixed on the first strand, monodromy factors $F_K$5 weight the remaining closure points, and $F_K$6.

Algorithmic pipeline

Phase 1 enumerates candidate multicycles, which encode inversion data as permutations on segment labels: deranged points receive negative signs, and "jumps" between over- and understrands realize the local patterns permitted by the nonzero $F_K$7-matrix cases. For homogeneous braids this phase is deterministic; otherwise candidates are generated over cyclic rotations and an optional horizontal mirror of the braid word, validated locally, and tested in parallel.

Phase 2 translates the definition of the admissible state set into a reduced integer linear constraint system: boundary conditions from the reduced trace, periodicity from braid closure, sign bounds, conservation equalities, and crossing-type inequalities ensuring nonnegative $F_K$8-binomial indices. A fixed-point reduction propagates constants, merges symmetric inequalities into equalities, and parametrizes all state variables affinely in a small set of free parameters. The HiGHS mixed-integer solver then certifies that the polyhedron, augmented with degree-truncation constraints, is nonempty and bounded — rejecting the candidate if any parameter direction is unbounded. Successful candidates are serialized to CSV.

Phase 3, a compiled C++ backend using FLINT polynomial arithmetic, enumerates integral points of the polytope by depth-first search with OpenMP parallelism over the first free parameter, evaluates the $F_K$9-matrix product per state (accumulating monomial prefactors as exponent offsets and expanding Pochhammer inverses as truncated geometric series), applies the reduced-trace normalization, and emits JSON with metadata including fractional power offsets.

A notable search-space reduction occurs in Phase 1: the naive space of sign assignments has size $\widehat{Z}$0 for $\widehat{Z}$1 crossings, but across 1245 nice knots with at most 12 crossings the number of locally admissible candidates never exceeded 310 (median 134) against a generic count of up to $\widehat{Z}$2 — a median reduction factor near $\widehat{Z}$3. Candidate search plus boundedness certification typically completes in well under a second.

Performance

Timings on a 12-core AMD Ryzen 9 7900X show that low-complexity knots are essentially instantaneous even at high truncation degree: the figure-eight knot reaches degree 50 in 1.6 s, while the trefoil at degree 10 takes under 0.1 s. Complexity grows sharply with braid length and degree: for the 12-crossing knot $\widehat{Z}$4, each increment of degree by 5 beyond degree 20 costs roughly an order of magnitude (1.1 s at degree 20, 13 s at 25, 116 s at 30). Parallel scaling is close to linear: the $\widehat{Z}$5 degree-25 computation takes 129 s single-threaded versus 13 s on 12 threads. Thread-safe memoization of $\widehat{Z}$6-binomials, Pochhammer factors, and per-crossing partial products contributes materially to throughput.

Validation

Correctness is checked against independent implementations and theoretical consistency conditions. The MMR relation supplies two analytic checks: the specialization $\widehat{Z}$7 must equal $\widehat{Z}$8, and the first $\widehat{Z}$9-derivative at $\widehat{Z}$0 must match a perturbed Alexander invariant $\widehat{Z}$1, cross-validated against Bar-Natan–van der Veen's polynomial-time diagrammatic algorithm, which is entirely independent of the $\widehat{Z}$2-matrix construction. For links, $\widehat{Z}$3 is compared against the iterated Laurent expansion of the inverse multivariable Alexander polynomial. Internal consistency tests verify agreement across different acceptable inversion data and across braid variants related by cyclic rotation or mirroring, consistent with the known independence of the inverted state sum from these choices.

Applications and mathematical consequences

The database generated with fkcompute enabled two results, some now theorems:

• Leading term: the coefficient of minimal $\widehat{Z}$4-power is a monomial (proved subsequently); for all knots up to 12 crossings its $\widehat{Z}$5-power equals $\widehat{Z}$6, the Seifert genus minus the Hopf invariant. This holds for arbitrary homogeneous braid knots but remains conjectural for general fibered knots.
• Slopes: the minimal powers of the coefficient Laurent polynomials form a quadratic quasipolynomial whose leading coefficient is conjecturally a boundary slope. Computing these slopes for fibered knots up to 10 crossings revealed knots such as $\widehat{Z}$7 with slope strictly below $\widehat{Z}$8, disproving the conjectured meromorphicity of Park's inverted Habiro series — a concrete negative result obtained directly from the computational data.

Limitations and open questions

The method applies only when an acceptable inversion datum exists for some braid variant of the input. This covers homogeneous braids and, conjecturally, all fibered knots, but existence is not guaranteed in general, and the package reports failure rather than falling back to another construction. Characterizing nice presentations, or developing a complementary algorithm for the remaining cases, is an open problem with direct practical impact. Moreover, Phase 1 explores only cyclic rotations and an optional mirror of the input word rather than the full Markov equivalence class, so a failing presentation may have an equivalent succeeding one. On the computational side, state counts grow rapidly with truncation degree and braid complexity, placing high-degree computations for long braids out of reach; the authors suggest smarter enumeration pruning, exploitation of polyhedron symmetries, and incremental extension of existing truncations as natural improvements.

Conclusion

fkcompute provides the first efficient, openly available implementation of the Gukov–Manolescu invariant, combining a permutation-based search for inversion data, ILP-based certification of finiteness, and a compiled FLINT-backed enumerator achieving near-linear parallel scaling. Its verified computations extend to prime knots of 12 crossings and links of 3 components, and the accompanying database has already yielded new theorems and a disproof of a meromorphicity conjecture. The principal open questions concern the scope of the nice-knot class and the reduction of Phase 3's exponential growth in degree and braid length.