Knot Theory Educator

What it does

When your agent uses it

• Creating interactive visualizations
• Explainer cards
• Step-wise animations
• Translating abstract algebra into intuitive understanding

Example prompts

• “/knot-theory-educator”

Requirements

• Pre-approved tools (allowed-tools): Read, Write, Edit, Bash, Glob, Grep

Workflow steps

4 steps, taken from the first numbered list in SKILL.md.

1. 1Start with physical demonstration (hold three strings)
2. 2Name the basic moves (left-over-middle = σ₁)
3. 3Show why certain moves can be reordered
4. 4THEN introduce formal notation as shorthand

What it can do on your machine

Read from SKILL.md and the folder at commit 6713fc7. It shows what the files ask for, not the result of running them.

• Tool permissions

  Pre-approves these tools, so the agent can use them without asking each time:

  • Read
  • Write
  • Edit
  • Bash
  • Glob
  • Grep

  From allowed-tools in the SKILL.md frontmatter.

• Runs code

  No scripts in the folder and no shell commands in SKILL.md (its code samples are javascript and html).

  From the folder's file list and the shell code blocks in SKILL.md.

• Network

  No URLs in SKILL.md.

  From URLs in SKILL.md, links to its own repository left out.

• Credentials

  Names no API keys, tokens, secrets or passwords.

  From names ending in _API_KEY, _TOKEN, _SECRET, _KEY or _PASSWORD in SKILL.md.

Context cost

Knot Theory Educator loads about 2.6k tokens when it runs, and up to ~8.3k if it reads all its reference files. Until then it costs about 129 tokens; SKILL.md has 730 words of instructions outside code blocks.

Always · name and description, kept in context so the agent knows when to use it

~129

When it runs · the whole SKILL.md, loaded when a task matches

~2.6k

With references · SKILL.md plus every file in references/, read only if the agent opens them

~8.3k

Estimates: characters ÷ 4, the usual rule of thumb; real counts depend on the model's tokenizer. Scripts and assets cost tokens only if the agent reads them.

Safety

• NotePre-approves every shell command (allowed-tools: Bash)SKILL.md

  allowed-tools: Read, Write, Edit, Bash, Glob, Grep

Automated static check — not a guarantee. Review scripts before installing. It scans the text of SKILL.md for risky patterns (piping downloads into a shell, reading credential files, hidden Unicode, destructive commands); files beside SKILL.md are not scanned.

SKILL.md

The full file from curiositech/some_claude_skills at commit 6713fc7, republished under its MIT licence (© curiositech). 730 words, ~2,591 tokens.

name

knot-theory-educator

description

Expert in visualizing and explaining braid theory, knot mathematics, and topological concepts for educational purposes. Use for creating interactive visualizations, explainer cards, step-wise animations, and translating abstract algebra into intuitive understanding. Activate on keywords: braid theory, knot visualization, σ notation, crossing diagrams, Yang-Baxter, topological education. NOT for general math tutoring, pure knot invariant computation, or non-educational knot theory research.

allowed-tools

Read, Write, Edit, Bash, Glob, Grep

metadata.category

Content & Writing

metadata.tags

knots, topology, braid-theory, visualization, education

Novice approach: "σ₁ is a generator of B₃ satisfying..."
Expert approach: "Imagine holding three strings. σ₁ means 'cross the
                  left string OVER the middle one.' Now they've swapped
                  positions. σ₁⁻¹? Cross it back UNDER."

  1   2   3          2   1   3
  │   │   │          │   │   │
  │ ╲ │   │    →     │   │   │
  │   ╳   │          │   │   │
  │ ╱ │   │          │   │   │
  │   │   │          │   │   │

  1   2   3          1   3   2
  │   │   │          │   │   │
  │   │ ╲ │    →     │   │   │
  │   ╳   │          │   │   │
  │   │ ╱ │          │   │   │
  │   │   │          │   │   │

<div class="explainer-card graph-paper">
  <h3>The Word Problem</h3>
  <p class="intuition">
    "Given two different-looking recipes for tangling strings,
    do they produce the same tangle?"
  </p>
  <p class="formal">
    Formally: Given braid words w₁ and w₂, determine if they
    represent the same element of Bₙ.
  </p>
  <p class="example">
    Example: Is σ₁σ₂σ₁ the same as σ₂σ₁σ₂? (Yes! Yang-Baxter)
  </p>
  <p class="complexity">
    Solved by Artin (1947) - polynomial time in word length
  </p>
</div>

// Animation sequence for σ₁σ₂σ₁⁻¹
const steps = [
  { state: 'initial', label: 'Three untangled strands: ε (identity)' },
  { state: 'after_s1', label: 'σ₁: Left crosses over middle', highlight: [0,1] },
  { state: 'after_s2', label: 'σ₂: Middle crosses over right', highlight: [1,2] },
  { state: 'after_s1_inv', label: 'σ₁⁻¹: Left crosses UNDER middle', highlight: [0,1] },
  { state: 'final', label: 'Result: Strands repositioned, complexity = 3' }
];

┌─────────────────────┬─────────────────────┐
│   TWO STRANDS (B₂)  │  THREE STRANDS (B₃) │
├─────────────────────┼─────────────────────┤
│ One generator: σ₁   │ Two generators: σ₁,σ₂│
│                     │                     │
│ Abelian (order      │ NON-abelian         │
│ doesn't matter)     │ (order MATTERS!)    │
│                     │                     │
│ σ₁σ₁⁻¹ = ε always  │ σ₁σ₂ ≠ σ₂σ₁        │
│                     │                     │
│ Always untangle by  │ May need complex    │
│ counting crossings  │ algorithms to solve │
│                     │                     │
│ Like a single dial  │ Like a Rubik's cube │
└─────────────────────┴─────────────────────┘

t=0:  R───G───B  (initial positions)
t=1:  G───R───B  (after σ₁: R crossed over G)
t=2:  G───B───R  (after σ₂: R crossed over B)

// Simplified physics demo with thick rope rendering
class BraidDemo {
  constructor(numStrands) {
    this.strands = numStrands;
    this.crossings = [];
    this.mode = 'interactive'; // or 'playback'
  }

  // Render thick ropes with clear over/under
  renderThickRope(strand, ctx) {
    ctx.lineWidth = 20;
    ctx.lineCap = 'round';
    // Draw shadow pass first (creates depth)
    // Then main strand with gradient
  }

  // Highlight the key insight
  showComplexityDifference() {
    if (this.strands === 2) {
      return "Count crossings. Apply that many σ₁⁻¹. Done.";
    } else {
      return "Must track which strand crossed which. Order matters!";
    }
  }
}

Is the concept about static structure or dynamic process?
├── Static (e.g., "what is a braid group?")
│   └── Use: Comparison cards, diagrams with annotations
└── Dynamic (e.g., "how does σ₁ work?")
    ├── Is it a single operation?
    │   └── Use: Before/after with animation between
    └── Is it a sequence?
        └── Use: Step-wise timeline with scrubbing

© curiositech, MIT. Rendered from Markdown: HTML in the file is shown as text, images as links, and headings moved down two levels. Raw file

Files

SKILL.md and 2 other files (references) in .claude/skills/knot-theory-educator of curiositech/some_claude_skills.

• SKILL.md
• .claude-plugin/plugin.json
• references/knot-garden-education.md

Open the folder on GitHubat commit 6713fc7