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arxiv: 2605.14189 · v1 · submitted 2026-05-13 · 🧮 math.GT

Recognition: 1 theorem link

· Lean Theorem

The KnotMosaics Package for SageMath

Mary Y. Deng , Allison K. Henrich , Sean H. Kawano , Andrew R. Tawfeek
Authors on Pith no claims yet

Pith reviewed 2026-05-15 01:32 UTC · model grok-4.3

classification 🧮 math.GT
keywords knot mosaicsSageMathplanar diagramsknot invariantstangle mosaicscomputational topologyknot theory
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The pith

KnotMosaics package represents knot mosaics as matrices of tile labels to validate and analyze them in SageMath.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper introduces the KnotMosaics package for SageMath, which stores each n-mosaic as a matrix using standard tile labels. It applies local connectivity rules to validate the mosaic, trace strands and components, compute planar diagram codes, generate random examples, and construct rational tangle mosaics. The planar diagram interface links these representations to existing knot software, supporting calculations such as Jones polynomials and knot Floer homology checks. A sympathetic reader would care because the work turns a diagrammatic area of knot theory into programmable operations that support systematic exploration.

Core claim

The package implements knot mosaics by representing an n-mosaic as a matrix of standard tile labels and enforcing the local connectivity rules needed to validate mosaics, trace strands and components, compute planar diagram codes, generate random examples, and construct rational tangle mosaics. This representation connects directly to existing knot and link software in SageMath, enabling further computations such as Jones polynomials and knot Floer homology checks.

What carries the argument

Matrix of standard tile labels for an n-mosaic together with the local connectivity rules that validate the diagram and permit strand tracing and component identification.

If this is right

  • Random examples of knot mosaics can be generated automatically for further study.
  • Rational tangle mosaics can be constructed using the package's dedicated routines.
  • Planar diagram codes derived from mosaics enable direct computation of knot invariants such as the Jones polynomial.
  • Knot Floer homology checks become available for links obtained from mosaic diagrams.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The package could support systematic enumeration of all valid mosaics up to small sizes.
  • Users might apply it to test conjectures on the mosaic number of specific knots through computation.
  • Integration with SageMath could encourage more researchers to run experiments on mosaic representations rather than hand-drawn diagrams.

Load-bearing premise

The chosen tile labels and local connectivity rules correctly reproduce the standard mathematical definition of knot mosaics.

What would settle it

A concrete mosaic diagram that the package accepts as valid but which fails to satisfy the connectivity conditions in the established knot mosaic literature, or vice versa.

Figures

Figures reproduced from arXiv: 2605.14189 by Allison K. Henrich, Andrew R. Tawfeek, Mary Y. Deng, Sean H. Kawano.

Figure 1
Figure 1. Figure 1: The eleven standard mosaic tiles. Since their introduction, knot mosaics have attracted significant interest in combinatorial knot theory. Researchers have studied the mosaic number of a knot—the minimum n such that it can be represented on an n-mosaic—establishing bounds and tabulations for various knot families [9, 12, 10, 1]. The enumeration of knot n-mosaics and upper bounds for toroidal mosaics have a… view at source ↗
Figure 2
Figure 2. Figure 2: Four isotopic knot mosaic representations of the 52 knot, as de￾tected by the is isotopic function. By the Lomonaco–Kauffman conjecture, all these mosaics are related by a finite sequence of mosaic Reidemeister moves. This can be confirmed by comparing the corresponding SageMath links for any pair of mosaics M and N under the is isotopic method: sage: Link(M.pd_code()).is_isotopic(Link(N.pd_code())) True 4… view at source ↗
Figure 3
Figure 3. Figure 3: The zoom map applied to a mosaic, replacing each tile by an isotopy-equivalent 3 × 3 block. 4.4. Rational tangles. The package supports construction of rational tangle mosaics, fol￾lowing Conway’s algebraic tangle notation [3]. Positive integers produce mosaics using T10 crossings, negative integers use T9 crossings, and the special values 0 and ∞ produce single￾tile tangles. 0 1 2 2¯3 2¯32 ∞ −1 = ¯1 21 21… view at source ↗
Figure 4
Figure 4. Figure 4: Left: Conway’s original tangle examples from [3, [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗
read the original abstract

We introduce KnotMosaics, a SageMath package for constructing, visualizing, and analyzing knot mosaic diagrams. The package represents an n-mosaic as a matrix of standard tile labels and implements the local connectivity rules needed to validate mosaics, trace strands and components, compute planar diagram codes, generate random examples, and construct rational tangle mosaics. The planar diagram interface connects the mosaic representation to existing knot and link software, enabling computations such as Jones polynomials and knot Floer homology checks. We describe the package design, its main algorithms, and representative examples that illustrate how KnotMosaics can support computational exploration in knot mosaic theory.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 3 minor

Summary. The paper introduces the KnotMosaics package for SageMath. It represents an n-mosaic as a matrix of standard tile labels and implements the local connectivity rules needed to validate mosaics, trace strands and components, compute planar diagram codes, generate random examples, and construct rational tangle mosaics. The planar diagram interface connects the mosaic representation to existing knot and link software, enabling computations such as Jones polynomials and knot Floer homology checks. The manuscript describes the package design, its main algorithms, and representative examples.

Significance. If the implementation faithfully matches the standard mosaic definitions in the literature, the package provides a useful computational tool for knot mosaic theory. It bridges mosaic diagrams with established invariant calculators in SageMath, supporting exploration, verification, and potentially new computational experiments in the field. The representative examples serve as an empirical check on core operations.

major comments (2)
  1. §3 (Main algorithms): The local connectivity rules and strand-tracing procedure are described at a high level without explicit enumeration or pseudocode. Since these rules are load-bearing for the claim of faithful reproduction of the mathematical definition, they should be listed in full or directly referenced to the literature so readers can verify correctness independently of the source code.
  2. §4 (Examples): The random-mosaic and rational-tangle examples illustrate functionality but do not report a side-by-side comparison of an invariant (e.g., Jones polynomial) computed from the mosaic PD code versus the same knot given by a standard diagram; such a check would directly test the end-to-end correctness of the PD-code extraction step.
minor comments (3)
  1. Abstract and §2: The phrase 'knot Floer homology checks' should specify the exact SageMath or external routine used, as the interface is a key claimed feature.
  2. References: Add citations to the foundational mosaic literature (e.g., the original tile-set definitions) so the chosen labels and rules can be cross-checked against the source definitions.
  3. Installation and reproducibility: Include a direct link to the package repository or a one-line installation command; this is standard for software papers and aids immediate testing.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments on the manuscript. We address each major comment below.

read point-by-point responses

  1. Referee: §3 (Main algorithms): The local connectivity rules and strand-tracing procedure are described at a high level without explicit enumeration or pseudocode. Since these rules are load-bearing for the claim of faithful reproduction of the mathematical definition, they should be listed in full or directly referenced to the literature so readers can verify correctness independently of the source code.

    Authors: We agree that greater explicitness would strengthen verifiability. In the revised manuscript we will add a complete enumeration of the local connectivity rules (following the standard mosaic definitions) together with pseudocode for the strand-tracing procedure in §3, allowing readers to check correctness without inspecting the source code. revision: yes

  2. Referee: §4 (Examples): The random-mosaic and rational-tangle examples illustrate functionality but do not report a side-by-side comparison of an invariant (e.g., Jones polynomial) computed from the mosaic PD code versus the same knot given by a standard diagram; such a check would directly test the end-to-end correctness of the PD-code extraction step.

    Authors: We accept this suggestion as a useful empirical check. The revised §4 will include a side-by-side comparison of invariants (e.g., Jones polynomial) obtained from the mosaic-derived PD codes against the same knots presented by standard diagrams. revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper describes a software package implementing the standard n-mosaic definition (matrix of tile labels plus local connectivity rules) taken directly from the existing literature, along with standard operations such as strand tracing, PD-code extraction, and rational tangle construction. No new mathematical derivations, fitted parameters, predictions, or self-referential claims are asserted; correctness reduces to faithful encoding of external definitions, which is checked via representative examples and interfaces to existing knot software. The derivation chain is therefore self-contained with no reductions to the package's own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The package adds no new mathematical content; it encodes existing knot-mosaic definitions in software.

axioms (1)
  • domain assumption Standard tile labels and local connectivity rules for knot mosaics as defined in the knot theory literature
    The implementation rests on these pre-existing definitions without modification or proof.

pith-pipeline@v0.9.0 · 5407 in / 1112 out tokens · 62793 ms · 2026-05-15T01:32:27.638642+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

16 extracted references · 16 canonical work pages

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