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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

Measuring knots

A compact ball made from one closed strand of wool — Is this simplified enough?

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.

Definition.

The $K \# J$ of oriented knots $K$ and $J$ is formed by removing a small unknotted arc from each and joining the ends compatibly.

Theorem ().

Every nontrivial oriented knot is a connected sum of prime knots, uniquely up to order and 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 $\mathbb{S}^3$

1952

, : triangulation theorem
2003

Jaco, Rubinstein:
2006

Jaco, Rubinstein:
2007

Matveev:
2014

Burton:
2014

Andersen, Kashaev:
2023

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 $c\ge 5$.
2024

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!!
2025

He, Sedgwick, Spreer:

Factorises a knot given as an edge loop in a triangulation of $\mathbb{S}^3$ into its prime summands.
Before this project: $k$ trefoils cost at least $2k$ and at most $33k-1$ tetrahedra.
2026

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

Conjecture. $\tau(\#^{k}K_{3_1})=4k-3$.
Bounds for every knot.
How small can simplicial spheres with be?
…

Development

Source code languages

Software and resources

Words from the developer

Stormy Forest

I'm a crazy lover of vanilla JavaScript and Three.js. In short, I really love JS!! What attracts me most is how extraordinarily lightweight and compatible it is, along with the unparalleled interactivity it makes possible. Naturally, I'm just as fond of web and front-end design, and of the tools that come with them. I also produce figures and animations with TikZ/PGF, Swift, Asymptote, MATLAB, R, Desmos (JS) and Python (Matplotlib and Manim).

Stormy Forest is a collection of visualisation tools for topologists at different levels. I'm extremely grateful to the contributions from generations of brilliant open-source maintainers behind the software and resources listed above.

Gallery (not fully named, mini version in the slide)

2024 Spring

Pachner Move
2025 Fall

Triangulations and Legos
2025 Spring

Grids, Knots and Stamps

Note

This was supposed to be part of my SCDL3991, but sadly it was taken away from me by someone else (this is not a criticism of them, and I think they ended up doing a wonderful job). That said, I didn't give up on developing the software. There are still numerous conjectures about , and everybody is welcome to join the fun of exploring these masterpieces.
2026 Summer

Farey Tessellator
2026 Fall
- Slide deck
- Transition functions
Note

Built in DATA3888 Data Science Capstone.
2026 Winter
- Connected sum and slicer
- Skill: transition
2026 Spring
- Normalflächer
- Skill: navigation

Anecdote

The Start

I started Stormy Forest in 2024, while taking a Geometry and Topology course taught by Prof Stephan 'Tillus' Tillmann and Dr Grace Garden. Special thanks to Tillus for joining so many conversations over the years. He is so insightful, and always comes up with elegant figures on the first sketch. I wish I could be that good one day.

In 2025, during my SCDL3991 project with A/Prof Jonathan Spreer, I explored some other tools more systematically, and some notes on them are recorded in the project report. Jonathan encouraged me (twice) to pursue this officially as a new project, yet I never really claimed to be working on it, despite a few secret attempts.
The Bottle and The Neck

Not all those who wander are lost.

— J. R. R. Tolkien,

Working through the Gallery above has been one long process of overcoming bottlenecks. Algorithms for drawing knots are arguably mature, but this is absolutely not the case for triangulations, which come with limitations of their own in 3D space. Normalflächer is thus exceptionally hard. I wouldn't even say it has a bottleneck, since it is more or less a neck through and through. I have never been able to touch, or even to foresee, what the "bottle" actually is.
JS

Success is $80\%$ methodology, and $20\%$ persistence.

— Jonathan Spreer, MATH1964 office hour, 19 September 2024

My passion for JavaScript has already been mentioned. But there is another, very important JS in my studies – Jonathan. The one who proposed the figure-production project has long forgotten that it ever existed, but I haven't. All along I have hoped that beautiful ideas could be communicated well, at least well enough that no one would question the reason behind my endeavours.

When we were writing up the , I was entrusted with producing the figures. The moment I put in some (TikZed, far from perfected) figures, the first comment I received was "Where did you find these figures?" The second comment (after I complained) was "It's not conceivable to me how undergrad students can make these elaborated figures".

I have been turned into such a joke. Maybe I need to admit that I'm too mediocre for anyone to actually entrust me with a challenging task; maybe I do lack "persistence" after so much procrastination... I cry every time, just thinking about it. All I wished for was some properly designed figures in a paper with both our names on it, so that I wouldn't disgrace anyone. It's not conceivable to me what actually went wrong.

Failure is understandably a familiar melody to many researchers, but I was so eager not to be the reason for one. Yet this is another task I have failed at. If I could go back to the morning of 19 September 2024, I would never walk into that office and join that conversation about becoming a successful mathematician. Maybe I'm the last person in the world who still remembers that sentence, and still follows it as a motto. Nobody is waiting for me.

As of today, I have no excuses, but am just sincerely sorry that the mathematician I like and respect most is not at all proud of me. Being thrown away as a trouble maker, is the destiny I deserved.
The End

Man will ja immer diesen maximalen Erfolg haben, und am Ende, wenn ich elf Jahre zurückblicke, haben wir es halt einmal geschafft, und das ohne Fans.

— Joshua Kimmich,

25 September 2026, the day before the mid-semester break. It was the day I finally understood that all of my dreams were already gone.

In the hallway of Carslaw, in the dark, the old memories came back. I remember kneeling on the frozen ground at 13, because I was made to. I remember my mom saying I shouldn't take another degree in Australia, because that would be the end of my academic life, if it had ever existed. I remember my dad saying I should not pursue a PhD in the US, because mediocre people like me would never graduate.

Other people become honours students on projects they love. Other people go on to a PhD straight after honours. For them, it can take little more than asking. Why are these people never advised to try other things, or told that their toolbox is not yet enough? At the very least they are never abandoned, even without endeavouring at all.

I cried for five hours straight, and it almost killed me. There was no time to mourn. I gathered what strength I had left, dragged Normalflächer out of my dusty project folder, and went back to its bottlenecks, one after another.

I wept as I typed in Fisher Library. It was raining, and for the first time in my years at uni, the whole area had emptied before 11 pm. At 1 am, I rendered a version in which a glued normal surface could be drawn out of the tetrahedron. I looked around. No one was there.

Then the semester break came. A piece of code refused to compile after five rounds of debugging, and I broke down in tears. They kept rolling down my face. The girl sitting across the table quietly reached over and handed me a pack of Haribo.

I had pretty much given up looking for another supervisor for honours. I was thinking of quitting uni right after this. I caused Alex so much trouble, because he ended up being the only person still trying to talk things through and find me a way out.

Joshua Kimmich at least won the Champions League once, and he is Germany's captain. How many players in the world get to do something like that?
The Barber

I went to the barber on 29 September. The barber was delighted about the semester break at USYD: their day was fully booked, with so many radiant young people coming in, all looking forward to trips to Perth, to Darwin, to New Zealand, to all over the world. They sighed at how rich these people were (of course they are, who else would spend $100 on their hair??), then pointed at the bright blue sky outside and said, admiringly:

This is the best season of the year.

Then they asked me where I was travelling to. I said I was going back to Fisher Library to work. My bangs had grown so long that I couldn't properly see the screen. The barber had no idea of what happened to me, but they stayed silent from then on.

Afterwards, I messaged my friend, who had flown home for the holidays the day before, and told them the story. And here was their answer:

A book holds a house of gold.

— Emperor Zhenzong of Song,

They know me too well.

But what am I doing after all?

I don't work for money (I spend money to work), nor for admiration – I have already lost those. Nor am I fighting for a bright future, which I probably never had. And I don't get to speak at big conferences in front of a whole bunch of people either.

So that day, I changed my mind.

I went down to the UTS Library instead.
Fisher Library

Villainy is the passport for villains, nobleness is the epitaph for the noble.

— Bei Dao,

I have no good words for this library, despite its fame and hospitality. Probably because of all the lonely nights I have spent there.

On 15 October, some very diligent second-year students knocked on my desk, asking for help with their Real Analysis. Having seen the knots on my screen for so long, they were sure I could prove their lemma – which I knew was not true at all. Half an hour later, I had only confused myself, and found that the lemma itself was wrong (in all dimensions). (Told them I couldn't do it!) They said they didn't want to use AI to breeze through it, whilst I was imagining the pleasant lives of those who do. How sad that is.

Maybe, in future, people should avoid working, and especially researching, on what AI is particularly good at. If we truly want to do something, we should do it now. Again, nobody is waiting.

For years, I've sat in the same silent area on Level 4, working on all kinds of things. The past six weeks have been more than extreme; my eyes have been bloodshot since 25 September. Nobody has discovered the secret.

It's not just the bangs blurring my eyesight.
$11\frac{3}{4}$

If you have read this far, as a gesture of gratitude I'd like to share with you a fancy hidden feature on this slide. Press B, and swipe across the canvas from the left-most to the right-most and hold.
$\mathcal{G\&T}$

The eulogy of humanity is the eulogy of courage.

— Hirohiko Araki,

I mean nothing to GT, but during the past few years GT means everything to me. Maybe I would go on to spend my whole life reminiscing and mourning my younger years as a junior Topologist. Maybe people in the future will continue to produce other wonderful, or mediocre, or garbage work. Maybe Topology is already dead, together with my younger self.

Long live GT, GT for Geometric Topology, and GT for the Grand Tour.

$\mathfrak{Mia\ San\ Mia}$

Lezhi Lin, $kiwinL(7, 1)$

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

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

$\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

[AK14]Jørgen Ellegaard Andersen and Rinat Kashaev, A TQFT from Quantum Teichmüller Theory, Communications in Mathematical Physics 330 (2014), 887–934.
[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.
[BBP⁺26]Benjamin A. Burton, Ryan Budney, William Pettersson, et al., Regina: Software for low-dimensional topology, https://regina-normal.github.io/, 1999–2026.
[Bin59]R. H. Bing, An alternative proof that 3-manifolds can be triangulated, Annals of Mathematics 69 (1959), no. 1, 37–65.
[BKR26]R. İnanç Baykur, Robion C. Kirby, and Daniel Ruberman (eds.), K3: A new problem list in low-dimensional topology, Mathematical Surveys and Monographs, vol. 295, American Mathematical Society, Providence, RI, 2026.
[BL13]Bruno Benedetti and Frank H. Lutz, Knots in Collapsible and Non-Collapsible Balls, The Electronic Journal of Combinatorics 20 (2013), no. 3, P31.
[Bra21]H. R. Brahana, Systems of Circuits on Two-Dimensional Manifolds, Annals of Mathematics 23 (1921), no. 2, 144–168.
[Bur13]Benjamin A. Burton, Computational topology with Regina: algorithms, heuristics and implementations, Geometry and Topology Down Under, Contemporary Mathematics, vol. 597, American Mathematical Society, Providence, RI, 2013, pp. 195–224.

References (continued)

[Bur14]Benjamin A. Burton, A New Approach to Crushing 3-Manifold Triangulations, Discrete & Computational Geometry 52 (2014), no. 1, 116–139.
[FP30]F. Frankl and L. Pontrjagin, Ein knotensatz mit anwendung auf die dimensionstheorie, Mathematische Annalen 102 (1930), 785–789.
[Hak61]Wolfgang Haken, Theorie der Normalflächen: Ein Isotopiekriterium für den Kreisknoten, Acta Mathematica 105 (1961), no. 3, 245–375.
[Hat23]Allen Hatcher, Notes on basic 3-manifold topology, Available at https://pi.math.cornell.edu/~hatcher/3M/3Mdownloads.html, 2023.
[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.
[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.

References (continued)

[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.
[Moi52]Edwin E. Moise, Affine structures in $3$-manifolds. V. The triangulation theorem and Hauptvermutung, Annals of Mathematics 56 (1952), 96–114.
[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.
[Rol03]Dale Rolfsen, Knots and links, vol. 346, AMS Chelsea Publishing, Providence, RI, 2003.
[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.
[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.
[Sch49]Horst Schubert, Die eindeutige zerlegbarkeit eines knotens in primknoten, Sitzungsberichte der Heidelberger Akademie der Wissenschaften, vol. 3, Springer Berlin Heidelberg, January 1949, pp. 55–105.
[Sei35]H. Seifert, Über das Geschlecht von Knoten, Mathematische Annalen 110 (1935), 571–592.

References (continued)

[Tai84]P. G. Tait, On knots. Part II, Transactions of the Royal Society of Edinburgh 32 (1884), no. 2, 327–342, Plate XLIV.
[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

[Ecm26]Ecma International, ECMAScript 2026 Language Specification, https://ecma-international.org/publications-and-standards/standards/ecma-262/, 2026, ECMA-262, 17th edition; JavaScript language.
[Khr26]WebGL API Registry, https://registry.khronos.org/webgl/, 2011–2026, Browser graphics API underlying the local Three.js renderers for interactive yarn, knots, LEGO bricks, triangulated beads, normal surfaces, and the turning book; supports mesh drawing, shaders, depth ordering, and transparency.
[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.
[WHA26]WHATWG, HTML Living Standard, https://html.spec.whatwg.org/multipage/, 2026, HTML, custom elements, Canvas, dialogs, and browser interaction.
[Wor26]World Wide Web Consortium, Web Standards: CSS, SVG, and MathML, https://www.w3.org/standards/, 2026, Slide layout, animation, vector artwork, and semantic mathematics.

Design & Inspiration

[App26]Apple Inc., SwiftUI, Software framework, https://developer.apple.com/swiftui/, 2019–2026.
[Ara87]Hirohiko Araki, JoJo’s Bizarre Adventure, Shueisha, 1987, Manga serialised in Weekly Shōnen Jump; Will A. Zeppeli’s line in Part 1, Phantom Blood, is quoted in the developer’s anecdotes.
[Bei78]Bei Dao, The Answer, Jintian (1978), no. 1, Chinese title Huida; reprinted in Shikan, 1979; quoted in translation in the developer’s anecdotes.
[Hal26]Halfbrick Studios, Fruit Ninja, Video game for iOS, https://apps.apple.com/app/fruit-ninja/id362949845, 2010–2026, The swipe and its blade trail: inspiration for the crushing blade in Method II.
[Men26]Jan Mendelin, Kapitän Kimmich, Documentary film, ZDF, https://www.zdf.de/dokus/kapitaen-kimmich-movie-100, June 2026, Joshua Kimmich is quoted in the developer’s anecdotes.
[Mon]Monash University, Brand book: Colours, https://www.monash.edu/brandbook/brand-elements/colours, Palette source for the bibliography citation-landing confetti: Monash blue, Heritage blue, Electric blue and Blueberry.
[PABL18]Samuel Poincloux, Mokhtar Adda-Bedia, and Frédéric Lechenault, Geometry and elasticity of a knitted fabric, Physical Review X 8 (2018), 021075.
[PT26]Hermès Paris and Les French Twins, We made appear the HERMES horse, YouTube video: https://www.youtube.com/watch?v=npFlC88jNbA&t=1s, 7 July 2026.
[Row97]J. K. Rowling, Harry Potter and the Philosopher’s Stone, Bloomsbury, 1997, Platform Nine and Three-Quarters: inspiration for the hidden fractional slide.
[SYD23]University of Sydney visual identity guidelines, February 2023.
[Tol54]J. R. R. Tolkien, The Fellowship of the Ring, George Allen & Unwin, 1954, From the poem “All that is gold does not glitter”, quoted in the developer’s anecdotes.
[UQ26]The University of Queensland Design System, https://design-system.uq.edu.au/, 2026.
[Zha]Zhao Heng (Emperor Zhenzong of Song), Exhortation to Learning, Chinese title Lixue pian; Northern Song dynasty, reigned 997–1022; quoted in translation in the developer’s anecdotes.

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.
[Nod26]Node.js contributors, Node.js: JavaScript Runtime, https://nodejs.org/en/about, 2009–2026, Runtime for HTML formatting, bibliography and equation preparation, renderer bundling, source-language measurement, asset fingerprinting, offline packaging, and automated tests.
[NPM26]npm: JavaScript Package Management, https://docs.npmjs.com/about-npm/, 2010–2026, Installs the runtime and build dependencies and invokes the project’s source preparation, local preview, test, packaging, and deployment commands.
[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.
[Wal26]Evan Wallace, esbuild: An Extremely Fast Bundler for the Web, https://esbuild.github.io/, 2020–2026, Version 0.28.1; bundles the maintained knot renderer, Three.js engine, and room-lighting environment into the local browser script reused by the knot, LEGO, tetrahedron, bead, and related three-dimensional scenes.

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.
[Clo26b], Wrangler: Cloudflare Developer CLI, https://developers.cloudflare.com/workers/wrangler/, 2019–2026, Version 4.126.0; runs local Pages, Functions, and KV previews with the configured loopback access bypass, bundles the private presenter functions, and deploys the curated public asset distribution to Cloudflare Pages.
[Sko26]Filip Skokan, jose: JavaScript Object Signing and Encryption, https://github.com/panva/jose, 2019–2026, Version 6.2.10; retrieves Cloudflare Access JSON Web Keys and verifies presenter JWT signatures, issuer, audience, and validity claims before the protected presenter endpoints use the authenticated identity for notes and avatar storage.

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.
[Pop26]Poppler developers, Poppler: A PDF Rendering Library, https://poppler.freedesktop.org/, 2005–2026, The pdfinfo tool reads the source book’s page count and pdftoppm rasterizes its atlas cells and readable page textures; pdftoppm also rasterizes and crops the historical Tait plate used by the knot artwork.

Typography & Typesetting

[Ame26]American Mathematical Society, AMS-LaTeX Supporting Software, https://ctan.org/pkg/amscls, 1990–2026, The amsalpha bibliography style supplies alphabetic citation labels, global entry ordering, and repeated-author formatting for the generated reference slides and full citation text.
[Art26]Artifex Software, Inc., Ghostscript, https://www.ghostscript.com/, 1988–2026, Provides PostScript support for dvisvgm and the exact bounding boxes exported by preview, keeping outlined equations aligned with their measured TeX dimensions.
[AUC26]Preview: Extracting LaTeX Fragments for Output, https://ctan.org/pkg/preview, 2001–2026, Places each typeset inline or display expression on a tight isolated preview page, providing the bounding box used by dvisvgm alongside the separately recorded TeX box width, height, and depth.
[Gie26]Martin Gieseking, dvisvgm: A DVI-to-SVG Converter, https://dvisvgm.de/, 2005–2026, Converts LaTeX-generated DVI equation pages into self-contained SVG glyph outlines, retaining preview bounding boxes for the cached inline and display mathematics and its exact placement.
[HMT26]Morten Høgholm, Lars Madsen, and The LaTeX Project, Mathtools: Mathematical Tools for LaTeX, https://ctan.org/pkg/mathtools, 2004–2026, Supplies mathematical commands and layout when the authored slide and bibliography expressions are compiled into the outlined equation cache.
[Kok26]Ron Kok, Temml: A TeX-to-MathML conversion library in JavaScript, https://github.com/ronkok/Temml, 2022–2026, Parses authored TeX delimiters into annotated semantic MathML for accessible mathematics; the live deck pairs that semantic representation with the separately prepared NewTX SVG glyph outlines, also retained in standalone export.
[LaT26]LaTeX: A Document Preparation System, https://www.latex-project.org/, 1983–2026, Compiles authored mathematical expressions with Libertinus text, mathtools, newtxmath, and preview; writes the DVI pages and TeX box dimensions used to prepare precisely positioned SVG equations.
[LCS26]Raph Levien, Cyreal, and Michael Sharpe, Inconsolatazi4: Inconsolata with TeX Support, https://ctan.org/pkg/inconsolata, 2013–2026, Inconsolatazi4 Regular font version 1.015; TeX package version 1.121; the bundled face supplies bibliography URLs and code-like reference text with the alternate letter and quotation features used by the report’s URL typography.
[Lib26]Libertinus Fonts, https://github.com/alerque/libertinus, 2012–2026, Version 7.051; bundled Sans faces supply slide text and controls, while Serif faces supply emphasis, tool names, citations, and reference lists; Libertinus text fonts also accompany NewTX mathematics in the TeX equation build.
[M⁺26]John MacFarlane et al., Pandoc: A Universal Document Converter, https://pandoc.org/, 2006–2026, Converts BibTeX-generated LaTeX citation labels and bibliography bodies through a JSON document representation into HTML, preserving formatted names, links, emphasis, and mathematical expressions for the reference slides and citation text.
[Pat26]Oren Patashnik, BibTeX, https://ctan.org/pkg/bibtex, 1985–2026, Formats the complete bibliography with amsalpha, establishing alphabetic citation labels, global entry order, names, punctuation, and repeated-author conventions; isolated entry formatting supplies full author names for citations.
[Sha26]Michael Sharpe, NewTX: Alternative uses of the TX fonts, with improved metrics, https://ctan.org/pkg/newtx, 2012–2026, Package version 1.756; newtxmath version 1.754; supplies the visible mathematical glyphs in the prebuilt SVG equation cache, using the Libertinus-compatible setting and measured TeX dimensions for consistent slide and standalone rendering.
[TeX26]TeX Live, https://tug.org/texlive/, 1996–2026, Provides the LaTeX, BibTeX, amsalpha, dvisvgm, font packages, and preview tools used for local equation and bibliography preparation; their generated assets are retained for builds that reuse the prepared content.
[Wil19]Lars G. Willighagen, Citation.js: A format-independent, modular bibliography tool for the browser and command line, PeerJ Computer Science 5 (2019), e214, Citation.js core and BibTeX plugin version 0.7.21; parses bibliography keyword metadata to route entries into the literature and technology sections and their subsections, while BibTeX supplies the displayed formatting and citation labels.

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`

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`

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

A Necklace of Trefoils

Measuring knots

Background of knots

Definition.

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

Definition.

Conjecture.

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

Aims towards A Necklace of Trefoils

Method I. Construction

Method II. Proof

Definition.

Results

Upper bound.

Discussions

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

Bounds for every knot.

How small can simplicial spheres with three-edge knots be?

…

Development

Welcome to Slide \(11\frac{3}{4}\) - Hidden Features

Keys

In the demonstrations

Thank you!

Acknowledgment

Gluing maps for the edge-ideal triangulation construction

01 Assemble the octahedron

02 Pair the boundary faces

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

References

References (continued)

References (continued)

References (continued)

Technology & Design

How small can simplicial spheres with be?

Welcome to - Hidden Features