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KnottedGraph: Scalable knotted-graph topology for scientific and mathematical discovery

Hakan Akgün, Xianquan Yan, Kehan Liu, Zhaoyun Chen, Ching Hua Lee

arXiv:2609.31152v1cs.MScs.CGcs.SCmath.GTphysics.comp-ph

TL;DR

Graph and homology-based summaries do not generally determine spatial embedding, while standard knot and link polynomials need extensions for branching graphs. KnottedGraph converts heterogeneous scientific representations into knotted graphs and scales exact Yamada-polynomial evaluation. It reproduces published formulas for graph families up to 500 crossings and introduces LLM-assisted discovery from computed invariant data.

  • Problem

    Graph connectivity and homological quantities do not generally determine a spatial graph’s ambient-isotopy class, and standard knot and link polynomials require extensions for branching graphs.

    Source: Introduction · p. 1

  • Method

    KnottedGraph constructs a common knotted-graph representation from heterogeneous scientific inputs, then projects, encodes and evaluates it using the Yamada polynomial.

    Source: Introduction · p. 1

  • Results

    At 500 crossings, evaluations took 0.12 s, 2.4 s and 5.0 s for Θ(n), C_n(∞+) and Θ_s(∞+), respectively, reproducing their published formulas.

    Source: Scalable Yamada polynomial evaluation… · p. 5

  • Takeaways & Limitations

    Computed exact invariants can support repeated analysis across scientific parameter spaces and generated graph families, as well as LLM-assisted mathematical discovery.

    Source: Introduction · p. 1

Abstract

from arXiv · show

Scientific data span heterogeneous structures, including coordinates, networks, surfaces, volumes and fields, yet their topology can be quantified within a common framework through graph connectivity, cycle structure, genus and spatial embedding. Graph- and homology-based summaries do not determine spatial embedding, while standard knot and link polynomials require extensions to accommodate branching graphs. Here, we introduce KnottedGraph, a computational framework that converts such scientific representations to knotted graphs that retain graph connectivity and spatial embedding together. It constructs projected diagrams and PD codes, enabling various topological analyses, including Yamada-polynomial evaluation for topological classification. For scalable exact evaluation, it combines partial resolutions that leave the same unresolved connections and optimizes their processing order; the resulting algorithm is verified against published topological invariants of knotted graphs with up to 500 crossings. This scalability enables us to introduce an LLM-assisted mathematical-discovery methodology, in which computational topological data generated across knotted-graph families are used to identify candidate closed-form formulas. With this approach, we identify analytical Yamada-polynomials for generic graph motif families exhibiting Abelian and non-Abelian word sequences. Together, these scalable capabilities make knotted-graph topology computationally accessible across scientific domains, enabling large-scale classification and introducing a route from topological data to LLM-assisted AI4Math discovery.

Introduction

Scientific systems require representations that preserve graph connectivity and spatial embedding, which existing graph and homological summaries do not generally determine. KnottedGraph integrates heterogeneous inputs, projected-diagram representations and scalable Yamada-polynomial evaluation, and applies computed invariants to LLM-assisted mathematical discovery.

Introduction

  • Across biology, materials science and physics, topology-sensitive representations must retain graph connectivity and spatial embedding, including edge over- and undercrossing information.The relevant systems include biomolecular backbones, vascular and neuronal networks, polymers, engineered networks, Hamiltonian-derived geometries and vector-flow curves.Source: Introduction · p. 1
  • Graph connectivity and homological quantities do not generally determine a spatial graph’s ambient-isotopy class, while standard knot and link polynomials require extensions for branching graphs.Knotted graphs retain connectivity and spatial embedding together, and associated invariants such as the Yamada polynomial can distinguish them.Source: Introduction · p. 1
  • KnottedGraph integrates heterogeneous inputs through spatial-graph construction where needed, projection with edge over- and undercrossings, PD encoding and Yamada-polynomial evaluation.Surfaces and volumes require constructing a spatial graph while preserving connectivity and three-dimensional edge geometry; handlebody–spine descriptions are one option.Source: Introduction · p. 1
  • Combining resolutions with the same unresolved connections and ordering operations to limit unresolved arc ends reproduces published formulas for spatial graphs with up to 500 crossings.The scalable evaluation supports repeated invariant calculations across scientific parameter spaces and generated graph families.Source: Introduction · p. 1
  • The LLM-assisted discovery method uses computed Yamada values to identify candidate family relations, convert them to symbolic forms and test additional family members.This targets interpretable algebraic structure across knotted-graph families, where traditional analytical derivations are generally family-specific.Source: Introduction · p. 1
  • Proposed future directions include broader volumetric topologies, additional knot and graph invariants, and discovery across structured families such as braids, tangles and recursive constructions.These directions connect scalable embedding-sensitive analysis with formula discovery from exact invariant data.Source: Introduction · pp. 1–2

Results

KnottedGraph constructs and validates spatial-graph representations, scales exact Yamada-polynomial evaluation to published families with up to 500 crossings, and uses exact invariant data to identify candidate laws for commuting and order-sensitive motif families.

Independent tests of knotted-graph construction, projection and topological invariant evaluation

  • Across all 4,400 handlebody constructions, input and graph-level recovery checks pass, and seed–output normalized Yamada polynomials agree.The result is scoped to the tested handlebody regime and numerical conditions.Source: Independent tests of knotted-graph… · p. 4
  • Accepted regular projections of each tested subcubic graph yield the same normalized Yamada polynomial, and calculations reproduce tested graph-polynomial identities and published values.The tests include tree, cycle, bouquet, theta and planar K4 relations.Source: Independent tests of knotted-graph… · pp. 4–5

Scalable Yamada polynomial evaluation across knotted graphs

LLM-assisted mathematical discovery from exact knotted-graph data

  • The discovery workflow generates exact polynomials, gives the LLM only a discovery subset, then tests retained symbolic formulas or transfer representations on held-out members.Agreement on the held-out set provides a finite out-of-sample test.Source: LLM-assisted mathematical discovery… · p. 5
  • Three repeated-motif families yield candidate general-m formulas with a shared three-channel transfer structure, and each inferred relation agrees with independently evaluated family members.The formulas have different coefficients, with a diagonal transfer operator associated with the repeated motif.Source: LLM-assisted mathematical discovery… · p. 5
  • For mixed words in three motifs, commuting transfer operators make the candidate invariant depend only on motif multiplicities, not motif order.The proposed closed form applies to arbitrary mixed words in the family.Source: LLM-assisted mathematical discovery… · pp. 5–6
  • In the fixed-graph pure-braid family, words with identical motif counts can have different normalized Yamada polynomials, so block order remains visible to the invariant.Finite exact data reconstruct a 15-dimensional transfer representation, with relations verified on short and held-out long words.Source: LLM-assisted mathematical discovery… · p. 7

Discussion

KnottedGraph links scalable, embedding-sensitive topological analysis with a route from exact invariant data to mathematical formula discovery. The discussion highlights applications across scientific structures and possible extensions to broader topologies, invariants, and recursively generated families.

Discussion

  • The framework connects three-dimensional scientific geometry to embedding-sensitive analysis and uses exact Yamada-polynomial data to identify candidate closed forms and transfer structures.The resulting commuting and noncommuting transfer descriptions capture count-dependent and order-sensitive family laws, respectively.Source: Discussion · p. 7
  • Compression bodies may require set-valued constructions such as the Yamada set because one graph cannot generally encode their boundary, attachment, and nesting information.The discussion motivates extending the framework beyond handlebodies.Source: Discussion · p. 7
  • Parameter-dependent analyses could organize biomolecular, polymer, vascular, neuronal, and engineered-network families into topological maps for identifying transitions and comparing structures.The proposed use applies when changes in spatial embedding carry information beyond abstract graph connectivity.Source: Discussion · pp. 7–8
  • Additional knot, link, and spatial-graph invariants could provide complementary classifications and independent exact datasets for comparing reconstructed algebraic structures.The discussion names the Alexander, Jones, and HOMFLY–PT polynomials and proposes braid words, tangles, satellite operations, and other recursive constructions for broader discovery.Source: Discussion · p. 8

Code and data availability

  • Source code, documentation, workflows, and supporting data are available online.The paper points to supplementary software-architecture and reproducibility details.Source: Code and data availability · p. 8

Supplementary Note 1 | From heterogeneous inputs to knotted graphs

KnottedGraph routes heterogeneous inputs into a common representation that retains available graph connectivity and three-dimensional edge geometry. For surface and volume inputs, it constructs candidate graph spines, selects topology across scales, and cleans the result before projection and invariant evaluation.

Supplementary Note 1.A | Heterogeneous inputs and the knotted-graph interface

  • The common knotted-graph representation retains graph connectivity and three-dimensional edge geometry across normalized inputs, with connected components and optional orientation also preserved.Curves and explicit graphs enter directly; surfaces, volumes and fields first undergo graph construction, while diagrams and PD codes enter at invariant evaluation.Source: Supplementary Note 1 | From… · p. 12, Table 1 · p. 13

Supplementary Note 1.C | Skeleton extraction and multiscale graph construction

  • For a volumetric regular neighborhood, the pipeline thins the volume, converts the skeleton to a graph and cleans it before projection and topological analysis.The recovered graph supports connectivity, cycle and geometric measurements, or projection for invariant evaluation.Source: Supplementary Note 1 | From… · pp. 14, 17, Figure 2 · p. 15
  • Cleanup operations have distinct scope: degree-two chain collapse preserves the represented edge, while leaf removal and short-edge contraction depend on the intended object and scale.Leaves may be retained when degree-one edges are part of the scientific object; short-edge contraction is user-controlled and requires geometric checks beyond its threshold.Source: Figure 2 · p. 15, Supplementary Note 1 | From… · p. 17
  • Because voxelization can split one physical junction into nearby branch voxels, candidate graphs are constructed across neighboring junction scales and topology is selected by recurrence.Consecutive clean isomorphic candidates favor the smaller-scale representative; otherwise, a valid isomorphism class must recur at least twice, or the direct zero-radius trace is retained.Source: Supplementary Note 1 | From… · p. 16
  • Cross-scale recurrence tests abstract graph topology, not spatial-embedding equivalence, and voxel-scale geometric artifacts may remain.A known maximum valence can strengthen selection when available; the thick-handlebody benchmark uses a maximum degree of three and tests through four voxel hops.Source: Supplementary Note 1 | From… · p. 16
  • The graph is projected to a diagram and encoded as a PD code for exact Yamada evaluation, while retaining its three-dimensional vertex positions and edge polylines before projection.Graph connectivity and cycle structure can also be analyzed without the projection step.Source: Supplementary Note 1 | From… · p. 17

Supplementary Note 2 | From knotted graphs to projected diagrams

KnottedGraph converts a fixed spatial embedding into a projection-specific diagram and PD code for exact invariant evaluation. It samples and checks candidate views, recovers crossings from the original three-dimensional geometry, and serializes graph vertices and crossings with their incidence and over/undercrossing data.

Supplementary Note 2 | From knotted graphs to projected diagrams

  • Projection changes the representation of a normalized knotted graph without changing its connectivity, and the accepted projection is retained with crossing structure and PD encoding.The procedure distinguishes graph vertices from projection crossings, rejects nongeneric views, and determines over/under information from the original three-dimensional geometry.Source: Supplementary Note 2 | From knotted… · p. 17

Supplementary Note 2.A | Sampling and selecting regular projections

  • Candidate views are ordered from fewest crossings upward, and full diagram construction proceeds only until a valid view is found; difficult minimum-crossing views may be compared with alternatives that reduce simultaneously unresolved arc ends.Invalid views are excluded rather than coerced into PD codes.Source: Supplementary Note 2 | From knotted… · p. 18, Table 2 · p. 18

Supplementary Note 2.B | Crossing detection and edge over/undercrossing determination

  • Crossing recovery detects intersections on stored edge polylines and records each crossing's position along both original edges, allowing later edge cuts without a second geometric search.Projected intersections with agreeing interpolated heights on nonincident edges are rejected as unresolved over/undercrossings.Source: Supplementary Note 2 | From knotted… · p. 18

Supplementary Note 2.C | Arc splitting, cyclic incidence and PD serialization

  • PD serialization cuts projected edges at ordered crossings, labels the resulting arcs, and records graph vertices with incident arc labels in counter-clockwise order.Crossing records order four incidences so the overpassing pair follows the convention used by the Yamada calculation.Source: Figure 3 · p. 19, Supplementary Note 2 | From knotted… · p. 19
  • Different generic views of the same spatial embedding can produce different crossing layouts, arc labels and projection-specific PD strings for exact invariant calculation.The numerical arc labels are arbitrary identifiers; incidence, cyclic order and over/undercrossing information are retained.Source: Supplementary Note 2 | From knotted… · p. 19, Figure 3 · p. 19

Supplementary Note 3 | Exact Yamada evaluation from PD codes

This note presents exact Yamada-polynomial evaluation from PD codes as a boundary-state computation that combines equivalent partial resolutions, with processing order affecting computational cost. It describes validation against published identities and volumetric recovery benchmarks, including structured diagrams with up to 500 crossings and 4,400 inverse-construction cases.

Supplementary Note 3 | Exact Yamada evaluation from PD codes

  • For subcubic spatial graphs, the normalized Yamada polynomial is projection-independent and an ambient-isotopy invariant, but equality does not establish equivalence.For higher-valence graphs, normalization alone may not remove dependence on local cyclic vertex arrangement; outputs require projection labels unless a common rigid-vertex structure is established.Source: Supplementary Note 3 | Exact Yamada… · p. 20
  • The library’s exact algorithm reproduces independently published Yamada formulas in a performance demonstration on structured diagrams containing up to 500 crossings.KnottedGraph computes the Yamada polynomial exactly for arbitrary supplied PD representations, including graphs containing higher-valence vertices.Source: Supplementary Note 3 | Exact Yamada… · p. 20

Supplementary Note 3.B | Compiling the PD code and removing redundancies

  • Before evaluation, KnottedGraph stores PD connectivity and cancels admissible Reidemeister-II bigons only when the same edge segment passes over at both crossings.Each valid cancellation removes two crossings while preserving the raw regular-isotopy Yamada polynomial, reducing the formal resolution space from 3^n to 3^(n−2).Source: Supplementary Note 3 | Exact Yamada… · p. 21

Supplementary Note 3.A | Local crossing relation and the formal resolution space

  • Each crossing has three resolution choices with weights Y, Y−1, and 1, yielding 3^n complete assignments for n crossings.The algorithm combines partial histories once they have identical connectivity among open arc ends, summing their exact polynomial contributions without changing the state sum.Source: Supplementary Note 3 | Exact Yamada… · pp. 22–23

Supplementary Note 3.C | Combining equivalent connectivity states in exact Yamada evaluation

  • The exact evaluation incrementally updates boundary connectivity and polynomial weights, merging states that future operations cannot distinguish.It incorporates crossing resolutions and the crossing-free graph contribution during processing rather than constructing and separately evaluating every complete resolved graph.Source: Supplementary Note 3 | Exact Yamada… · pp. 22–23
  • High-degree vertices are implemented as deterministic connection steps involving at most three arc ends, without changing the represented graph or its vertex factor.Helper connections are bookkeeping only, and exactly one step carries the original vertex's factor of −1.Source: Supplementary Note 3 | Exact Yamada… · p. 24

Supplementary Note 3 | Exact Yamada evaluation from PD codes

  • Processing order affects cost because simultaneous open arc ends determine the number of possible connectivity states, so the algorithm prioritizes closing existing connections.For sufficiently difficult diagrams, it tests alternative starting steps and retains an order only when it reduces the maximum simultaneous open ends by at least two.Source: Supplementary Note 3 | Exact Yamada… · pp. 24–25
  • When processing ends with no open arc ends, the accumulated exact state sum is the raw Yamada polynomial, and normalization is applied only afterward.Coefficients are first computed with checked 64-bit integers; if any exceed that range, the exact calculation is repeated with arbitrary-size integers.Source: Supplementary Note 3 | Exact Yamada… · p. 25

Supplementary Note 4 | Independent validation of knotted-graph construction, projection and Yamada evaluation

  • The implementation reproduces tested tree, cycle, bouquet and theta-graph identities, plus planar K4 and additional cubic-graph values [26, 32, 33].The Negami calculation is compared with its defining edge-subset polynomial, and its Yamada specialization with the direct graph calculation.Source: Supplementary Note 4 | Independent… · p. 25
  • Every displayed structured-family benchmark point agrees exactly with its literature expression, including diagrams with up to 500 crossings.This validates the exact algorithm at the tested scale, but does not imply that every diagram with the same crossing number has comparable cost.Source: Supplementary Note 4 | Independent… · p. 26
  • All 4,400 inverse-benchmark cases pass the stated input and graph-recovery checks, and all 4,400 seed–output normalized Yamada comparisons agree.Yamada preservation is recorded separately from recovery checks when both polynomial evaluations are available.Source: Supplementary Note 4 | Independent… · p. 27
  • The inverse-benchmark checks establish consistency, not proof that voxel inputs are manifolds or ambient-isotopic neighborhoods; Topoly timing data are right-censored, not paired speedups.The voxelized input must satisfy digital-homology and thickness conditions, and equal Yamada polynomials are not a complete embedding-equivalence certificate.Source: Supplementary Note 4 | Independent… · p. 27

Supplementary Note 5 | Parameter sweeps of volumetric geometries

The parameter-sweep interface repeatedly constructs knotted graphs from volumetric regions and records a selected topological or graph-based quantity. It supports finite-window momentum-space calculations and reuses spectra across energy or gap thresholds.

Parameter-sweep interface

  • The interface records a knotted graph and an analysis quantity for each volumetric region, with modular stages that can begin from the representation appropriate to the input.The recorded quantity may be a spatial-graph invariant, fixed-diagram polynomial, abstract graph summary, or unavailable result.Source: Supplementary Note 5 | Parameter… · p. 29

Momentum-space application

  • Finite-energy Fermi-volume scans organize finite-window calculations across two parameters, without identifying composition-driven band evolution with a specific Fermi-surface Lifshitz event.The construction uses the geometry-to-graph interface for such scans.Source: Supplementary Note 5 | Parameter… · p. 29

Analytic benchmark models

  • The benchmark interpolations connect selected endpoint knot and link types, including Hopf-to-trefoil, unknot-to-Solomon, and trefoil-to-cinquefoil paths.The Solomon endpoint uses c0 = 0.333, not exactly 1/3.Source: Supplementary Note 5 | Parameter… · p. 30

Sampling assumptions

  • The momentum-space sampling uses B_k = [-π, π] × [-π, π] × [0, π], with opposite faces not identified periodically, so boundary-touching inputs are finite-window objects.The full field includes both cos k_z and cos 2k_z, making [0, π] nonperiodic for that field.Source: Supplementary Note 5 | Parameter… · p. 30

Material scans

  • For material scans, diagonalizing once per sampled Hamiltonian parameter lets the calculation reuse band energies across energy or gap thresholds without storing all spectra.The corresponding graph outputs still require graph construction and equivalence checks for interpretation as source-topology labels.Source: Supplementary Note 5 | Parameter… · p. 30

Supplementary Note 6 | Geometric preconditioning of difficult embeddings

Geometric preconditioning reshapes sampled curves in three dimensions to make spatial-graph diagrams more readable while retaining their graph structure. Relaxation, projection, and invariant checks form a pipeline, but the objective does not guarantee a minimum-crossing projection.

Purpose and pipeline

  • Preconditioning moves edge curves while retaining graph vertices and incidence, addressing folded detours, close approaches, and bends that can obscure projected crossings.KnottedGraph maps the relaxed curves back to the graph for projection and invariant evaluation.Source: Supplementary Note 6 | Geometric… · p. 30

Projection and evaluation

  • The smoothed curves are projected afterward; over-under crossings are recorded in PD codes and evaluated with Yamada polynomials.The relaxation objective acts on three-dimensional curves, while projection selection occurs after relaxation.Source: Supplementary Note 6 | Geometric… · p. 31, Figure 9 · p. 32

Protein-derived examples

  • In the protein-derived theta graphs, unfolding, compaction, and smoothing respectively open close approaches, shorten detours, and regularize curves while preserving the three paths and shared endpoints.The selected theta graphs represent only chosen paths and bridges, not the complete molecular and crosslink networks.Source: Supplementary Note 6 | Geometric… · p. 31, Figure 9 · p. 32

Relaxation objective

  • The objective combines tangent-point repulsion with length, bending, and optional tube-clearance penalties, but because crossing number is absent, minimum-crossing projections are not guaranteed.The nonnegative weights set the emphasis of each term.Source: Supplementary Note 6 | Geometric… · pp. 30, 33

Topology checks

  • Trial updates are accepted only after motion checks and an Armijo energy-decrease test; rejected steps are halved and retried.Checks examine swept motions of nonincident segments, and contact tolerance is specified in input coordinate units.Source: Supplementary Note 6 | Geometric… · p. 33

Constraints and validation

  • Original graph vertices are pinned by default, but higher-valence junction geometry needs separate constraints and validation; invariant comparisons there require a common rigid-vertex structure.For subcubic graphs, normalized Yamada agreement supplies a consistency check, but polynomial equality alone does not establish isotopy.Source: Supplementary Note 6 | Geometric… · p. 33

Solver settings

Supplementary Note 7 | Inferring and testing Yamada family formulas from Yamada data

The paper treats exact Yamada-polynomial values as data for inferring candidate family formulas, rather than requiring a family-specific algebraic construction in advance. It reconstructs transfer descriptions for commuting and order-sensitive families and checks them against additional evaluations.

Relation to prior approaches

  • Earlier analytical family formulas rely on family-specific constructions or invariant relations, whereas this approach starts from exact values without prescribing that algebra in advance.The paper situates the method alongside existing Yamada-family and transfer-matrix derivations.Source: Supplementary Note 7 | Inferring and… · p. 34

Data-to-structure method

  • Exact Yamada values from selected family members are used to propose recurrences, closed forms, or transfer structures, which are fixed before additional members are evaluated as tests.This reverses the usual direction in which a known family construction supplies the analytical derivation.Source: Supplementary Note 7 | Inferring and… · p. 34

Commuting transfer laws

  • Three homogeneous motif families and one mixed family share a reconstructed three-channel transfer description, with different diagonal operators for their local motifs.The closed-form terms correspond to transfer channels propagated through repeated cells.Source: Supplementary Note 7 | Inferring and… · p. 35

Commuting mixed family

  • The mixed-family count-only formula agrees with direct evaluation on all 459 words in the completed plan and is invariant under motif reordering by construction.The finite comparisons support the proposed identification with the graph family.Source: Supplementary Note 7 | Inferring and… · p. 36

Order-sensitive family

  • For the fixed cubic graph with ordered braid words, the invariant distinguishes AAB, ABA, and BAA despite their identical motif counts (2, 1).The reconstructed transfer operators are noncommuting, retaining word-order information.Source: Supplementary Note 7 | Inferring and… · pp. 36–37

Out-of-basis prediction

  • The worked word AAA BA lies outside the 15 coordinate words, and its polynomial is obtained from the finite transfer representation using basis-word data.This example is not an independent holdout because its value is already present in the Hankel input.Source: Supplementary Note 7 | Inferring and… · pp. 36–38

Hankel reconstruction

  • The pure-braid transfer representation is reconstructed from Hankel data using words of length at most nine, with exact checks of rank, transfer relations, noncommutativity, and cubic identities.The finite block has rank fifteen; the argument notes that a global upper bound requires showing the local braid action and external closure factor through the space.Source: Supplementary Note 7 | Inferring and… · pp. 36–38

Verification

  • Transfer expressions agree coefficient-by-coefficient with direct evaluation on 255 distinct short words and 20 extrapolation cases of lengths 101, 125, 150, and 200.The longer cases are beyond the short-word reconstruction data.Source: Supplementary Note 7 | Inferring and… · p. 38

Supplementary Note 8 | Software architecture and reproducibility

KnottedGraph organizes its software around the objects passed between modules, converting heterogeneous inputs into a shared embedded-graph core. The architecture then supports projection and invariant evaluation, alongside layout, visualization, and scientific applications; reproducibility also requires calculation-specific records beyond the package itself.

Supplementary Note 8 | Software architecture and reproducibility

  • Modules are divided by the object they receive and return, allowing the same code to serve every example while applications remain outside the topology core.Application workflows call the shared extraction, projection, and invariant functions; adding an application does not change the topology implementation.Source: Supplementary Note 8 | Software… · pp. 38, 40
  • Curves and graphs pass directly to the core, whereas surfaces, volumes, and fields go through skeletonization and extraction, then graph cleanup.Cleanup can remove voxelization artifacts, but leaf removal and short-edge contraction are conditional because they can delete real features.Source: Supplementary Note 8 | Software… · pp. 38–39
  • The Embedded Graph Core stores vertex positions and ordered 3D edge polylines while preserving parallel edges and self-loops.The same graph can be projected, relaxed, drawn, or analyzed downstream.Source: Supplementary Note 8 | Software… · p. 39
  • Repulsive Layout changes graph coordinates without changing connectivity, and keeps vertices fixed by default.It is intended for valid embeddings that are difficult to project, such as when separate edges pass very close together.Source: Supplementary Note 8 | Software… · p. 39
  • Projection converts a 3D graph to a planar diagram and PD code for exact Yamada-polynomial evaluation.The calculation starts with 64-bit coefficients and automatically switches to arbitrary-size integers if a coefficient exceeds that range.Source: Supplementary Note 8 | Software… · p. 39
  • Reproducing a calculation requires its input data, preprocessing choices, parameters, implementation choices, and verification records, not just the package.The source revision, lockfile, and parameter record fix the software configuration; workflow-specific functionality is available as optional extras, and Python 3.11 or later is required.Source: Supplementary Note 8 | Software… · p. 40
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