A Necklace of Trefoils

Edge-ideal triangulations of the three-sphere realising \(k\)-fold connected sums of knots.

Presenter:Lezhi Lin

Supervisors:Dr. Alexander He and A/Prof Jonathan Spreer

The University of Sydney

Measuring knots

A compact ball made from one closed strand of wool

How knotted is this loop?

Background of knots

A smooth circular projection of the trivial knot
Trivial knot
A smooth trefoil diagram, with three crossings
Trefoil knot
A smooth figure-eight knot diagram, with four crossings
Figure-eight knot

Definition.

A \(K\) is a smoothly embedded circle \(\mathbb{S}^1\) \(\hookrightarrow\) \(\mathbb{S}^3\), up to ambient isotopy.

\(\mathbb{S}^1=\partial B^2\)
\(\mathbb{S}^2=\partial B^3\)
  • Closed
  • Connected
  • \({}=V-E+F=2\)

\(\mathbb{S}^3=\partial B^4\)

Measuring complexity in \(\mathbb{S}^3\)

Trefoil knot \(c(K_{3_1})=3\)
Ugly trivial knot \(c(U)=0\)
Trefoil knot \(c(K_{3_1}')=3\)

Definition.

The \(c(K)\) of a knot \(K\) is the minimum number of crossings among all its planar diagrams.

Conjecture.

Crossing number is : \(c(K\#J)=c(K)+c(J)\).

Edge-ideal triangulation of \(\mathbb{S}^3\)

  1. , : triangulation theorem

  2. Jaco, Rubinstein:

  3. Jaco, Rubinstein:

  4. Matveev:

  5. Burton:

  6. Andersen, Kashaev:

  7. Ben Aribi, Guéritaud, Piguet-Nakazawa:

    Realises every twist knot with \(c\) crossings as one edge of \(\lfloor c/2\rfloor+2\) tetrahedra, conjectured minimal for \(c\ge 5\).

  8. Ibarra, Mathews, Purcell, Spreer:

    Realises every knot as one edge of a one-vertex \(\mathbb{S}^3\).

    This paper (of very high quality) is recently accepted in JCTA (a very good journal), to appear in February 2027!!

  9. He, Sedgwick, Spreer:

    Factorises a knot given as an edge loop in a triangulation of \(\mathbb{S}^3\) into its prime summands.

  10. Before this project: \(k\) trefoils cost at least \(2k\) and at most \(33k-1\) tetrahedra.

  11. L., Spreer:

    Builds a canonical one-vertex \(\mathbb{S}^3\) for every torus knot.

Aims towards A Necklace of Trefoils

Edge-ideal triangulations of the three-sphere realising \(k\)-fold connected sums of trefoils.

Method I. Construction

Aim I.
Construct a nice combinatorial family of edge-ideal triangulations for every \(k\).

Method II. Proof

Aim I.
Construct a nice combinatorial family of edge-ideal triangulations for every \(k\).
Aim II.
Prove that the triangulation admits a decomposing \(\mathbb{S}^2\), and that the marked edge realises the \(k\)-fold connected sum of trefoils.

Definition.

A surface in \(\mathbb{S}^3\) is if it meets each tetrahedron of \(\mathcal{T}\) in disjoint discs, each a triangle cutting off one vertex or a quadrilateral separating two opposite edges.

4 types of Triangles. 3 types of Quadrilaterals.

Results

Aim I.
Construct a nice combinatorial family of edge-ideal triangulations for every \(k\).
Aim II.
Prove that the triangulation admits a decomposing \(\mathbb{S}^2\), and that the marked edge realises the \(k\)-fold connected sum of trefoils.
Aim III.
Tighten the upper and lower bounds on the complexity of edge-ideal triangulations.
An edge-ideal triangulation of \(\mathbb{S}^3\) carrying the \(k\)-fold connected sum of trefoils.

Upper bound.

\[\begin{aligned}\tau(\#^{k}K_{3_1})&\le 1+4(k-1)\\&=4k-3.\end{aligned}\]

Discussions

  1. Conjecture. \(\tau(\#^{k}K_{3_1})=4k-3\).

  2. Bounds for every knot.

  3. How small can simplicial spheres with be?

  4. …

Development

    Source code languages

    Welcome to - Hidden Features

    Keys

    1…9
    Jump to a slide; a lone digit waits a moment.
    ←↑PgUpBackspace
    Previous build, then the previous slide at its last build.
    →↓PgDnSpaceEnter
    Next build, then the next slide.
    B
    Method II, crushing layer: toggle the blade, then drag to cut.
    EscO
    Overview: arrows browse, Enter opens. Down beneath Development finds this slide.
    F
    Fullscreen.
    HomeEnd
    First and last slide.
    L
    Replay the current layer; reset its demo views.
    P
    Presenter window: notes, clock, next slide.
    R
    Back to the title; forgets the citation you left.
    ShiftF10
    Next build, like a right-click.
    Z
    Return to the citation you left.

    In the demonstrations

    +−0
    Zoom the selected element; Enter selects the next.
    A heading
    Open any aim or step on its own; Next continues from it.
    A keyword
    Jump to its bibliography entry, with a highlight or confetti; Z brings you back.
    Click a page
    Turn pages in the closing book (Background II).
    Click, then scroll
    Zoom a window on Aims or Methods.
    Double-click
    Stamp a knot flat or lift it; reset a demo view.
    Drag
    Turn a knot or solid; on Background I, drag a knot into another to sum them.
    Scroll or pinch
    Zoom a demo, Tait's table or the closing book; drag pans the table.
    Select two faces
    Pair the bead's faces yourself (Methods I, Step 2); a mismatch starts over.
    Shiftdrag
    Rotate only the knot (Background I); tilt a bead or the octahedron freely.
    Wheel in a pane
    Scroll the timeline, aims or steps; Up and Down scroll a focused pane first.

    Thank you!

    Presenter:Lezhi Lin

    Supervisors:Dr. Alexander He and A/Prof Jonathan Spreer

    The University of Sydney

    Acknowledgment

    • We acknowledge the Gadigal of the Eora Nation, the Traditional Custodians of the land on which stands, and pay our respects to Elders past and present.
    • This presentation is submitted in partial fulfilment of the assessment requirements for SCDL3992 Science Dalyell Individual Research Project at . I'm extremely grateful to my supervisor, Dr. Alexander He, for his patience and great insights. In acknowledgment of Alex, the normal surfaces and the crushing blade are coloured with the primary colour of . I'd also like to thank A/Prof Jonathan Spreer for hosting this project and for pointing me to the back in August 2025.
    • Generative AIs, including Codex 6 Astra and 6.1 Sol, Claude Opus 5.5, and Fable 5.1, were used to assist with this project. These models were especially helpful with implementing rapid navigation, maintaining citations and labels, debugging and testing. Legacy models were also used earlier in development. All outputs have been reviewed, edited, and validated.

    Gluing maps for the edge-ideal triangulation construction

    0123
    0123
    0123
    0123
    0123
    \(\mathcal{T}_0\)\(\mathcal{T}_1\)\(\mathcal{T}_2\)\(\mathcal{T}_3\)\(\mathcal{T}_{3_1}\) (solid torus)

    01 Assemble the octahedron

    Face Face Map
    \(\mathcal{T}_{3_1}.1\) \(\mathcal{T}_2.0\) 2031
    \(\mathcal{T}_{3_1}.2\) \(\mathcal{T}_3.0\) 3201
    \(\mathcal{T}_1.3\) \(\mathcal{T}_2.3\) 2103
    \(\mathcal{T}_1.2\) \(\mathcal{T}_3.2\) 3120

    02 Pair the boundary faces

    Face Face Map
    \(\mathcal{T}_{3_1}.0\) \(\mathcal{T}_{3_1}.3\) 3201
    \(\mathcal{T}_2.1\) \(\mathcal{T}_3.1\) 0132
    \(\mathcal{T}_2.2\) \(\mathcal{T}_3.3\) 0132

    03 Attach to \(\mathcal{T}_0\)

    Face Face Map
    \(\mathcal{T}_0.0\) \(\mathcal{T}_1.0\) 0321
    \(\mathcal{T}_0.3\) \(\mathcal{T}_1.1\) 2031

    Hover over a row to preview the paired faces.

    References

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    • [BAGPN23]Fathi Ben Aribi, François Guéritaud, and Eiichi Piguet-Nakazawa, Geometric triangulations and the Teichmüller TQFT volume conjecture for twist knots, Quantum Topology 14 (2023), no. 2, 285–406.
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    References (continued)

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    • [He25]Alexander He, AlexHe98/idealedge, 2025, Software artifact (2025 snapshot), swhId: swh:1:dir:ebcbe72831246d6d99ed7ff98f3cc00940ed808c.
    • [HSS25]Alexander He, Eric Sedgwick, and Jonathan Spreer, A Practical Algorithm for Knot Factorisation, 41st International Symposium on Computational Geometry (SoCG 2025) (Dagstuhl, Germany) (Oswin Aichholzer and Haitao Wang, eds.), Leibniz International Proceedings in Informatics (LIPIcs), vol. 332, Schloss Dagstuhl – Leibniz-Zentrum für Informatik, 2025, Full version, arXiv:2504.03942, pp. 55:1–55:15.

    References (continued)

    • [IMPS27]Dionne Ibarra, Daniel V. Mathews, Jessica S. Purcell, and Jonathan Spreer, Triangulations of the \(3\)-sphere with knotted edge, Journal of Combinatorial Theory, Series A 226 (2027), 106269, Published online 2026.
    • [JR03]William Jaco and J. Hyam Rubinstein, 0-efficient triangulations of 3-manifolds, Journal of Differential Geometry 65 (2003), no. 1, 61–168.
    • [JR06], Layered-triangulations of 3-manifolds, 2006, Preprint, arXiv:math/0603601.
    • [LS26]Lezhi Lin and Jonathan Spreer, Torus knots as loop-edges in three-sphere triangulations, 2026, Preprint, arXiv:2609.14200.
    • [Mat07]Sergei Matveev, Algorithmic Topology and Classification of \(3\)-Manifolds, 2 ed., Algorithms and Computation in Mathematics, vol. 9, Springer, Berlin, 2007.
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    • [OSS15]Peter S. Ozsváth, András I. Stipsicz, and Zoltán Szabó, Grid Homology for Knots and Links, Mathematical Surveys and Monographs, vol. 208, American Mathematical Society, Providence, RI, 2015.
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    • [Sav12]Nikolai Saveliev, Lectures on the topology of 3-manifolds: An introduction to the casson invariant, 2 ed., De Gruyter Textbook, De Gruyter, Berlin/Boston, 2012.

    References (continued)

    • [Sch11]Saul Schleimer, Sphere recognition lies in NP, Low-Dimensional and Symplectic Topology, Proceedings of Symposia in Pure Mathematics, vol. 82, American Mathematical Society, Providence, RI, 2011, pp. 183–213.
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    • [Whi49]J. H. C. Whitehead, Combinatorial homotopy. I, Bulletin of the American Mathematical Society 55 (1949), 213–245. MR 30759

    Technology & Design

    Browser & 3D Graphics
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    • [Thr26]Three.js authors, Three.js: JavaScript 3D library, https://threejs.org/, 2010–2026, Versions 0.128.0 (yarn demo) and 0.185.1 (shared scene engine); renders spatial knots and connected sums, LEGO brick examples, bead and necklace assembly, normal-surface tetrahedra, and the turning book, including meshes, cameras, materials, lighting, transparency, and projected shadows.
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    Design & Inspiration
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    Development & Build Tools
    • [Git26a]Git contributors, Git: Distributed Version Control, https://git-scm.com/, 2005–2026, Version control for authored slide sources, reusable scene engines, build scripts, and assets; the revision history supports reviewing changes and restoring earlier implementations.
    • [Git26b]GitHub, Inc., GitHub, https://github.com/about, 2008–2026, Hosting the source repository and its revision history, and providing the Git source for the Cloudflare Pages deployment workflow.
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    • [Pyt26]Python Software Foundation, Python, https://www.python.org/, 1991–2026, Runs the static no-cache preview server, including changed-source bibliography and equation refresh and source-language remeasurement; also prepares the closing-book page images and texture atlas through local image tools.
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    Hosting & Presenter Services
    • [Clo26a]Cloudflare, Inc., Cloudflare Developer Platform, https://developers.cloudflare.com/, 2017–2026, Pages hosts the public deck and deployment previews; Workers-based Pages Functions provide the private presenter API, Access protects presenter login, and Workers KV stores per-presenter notes and uploaded avatar settings in separate production and preview namespaces.
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    Image Processing
    • [Pil26]Pillow contributors, Pillow: The Python Imaging Library, https://python-pillow.org/, 2010–2026, Converts rendered PDF pages into grayscale JPEGs, resizes and pastes page cells into the closing-book texture atlas, and writes the optimized full-page textures loaded during the page-turn animation.
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    Typography & Typesetting
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