Knots, links, and the fundamental Reidemeister moves
A mathematical knot is a closed loop in 3D space with no loose ends. The big question is deceptively simple: when are two tangled loops secretly the same knot?
Select a knot and watch it turn in 3D. Try to imagine untying the trefoil without cutting it; it can't be done.
Knot theory is the mathematical study of knots -- closed loops embedded in 3D space. Unlike the knots you tie in shoelaces (which have loose ends), mathematical knots are closed curves with no endpoints. The fundamental question in knot theory is: when are two knots equivalent?
Two knots are considered the same if one can be continuously deformed into the other without cutting the rope or passing it through itself. This seemingly simple question leads to deep mathematics and surprising applications in biology, chemistry, and physics.
A knot is a closed curve embedded in 3D space. Explore famous knots and see how they differ in their 3D structure. Knots are considered equivalent if one can be continuously deformed into another without cutting the rope.
Key insight: Two knots are equivalent if one can be continuously deformed into the other without cutting the rope or passing it through itself. The trefoil knot cannot be unknotted -- it is fundamentally different from the unknot.
Select a knot to see it flattened into a diagram. At each crossing, the gap in the line shows which strand passes under.
The simplest non-trivial knot with 3 crossings.
A knot diagram is a 2D projection of a knot showing which strand passes over or under at each crossing. The diagram records the essential information about the knot's structure.
Key insight: The crossing number is the minimum number of crossings in any diagram of the knot. It is a topological invariant -- no matter how you deform the knot, you cannot reduce the number of crossings below this minimum.
Select a move to see the twist, poke, or slide. Each one changes the drawing but never the knot.
You can add or remove a simple twist without changing the knot.
The Reidemeister moves are three types of local changes you can make to a knot diagram without changing the knot itself. Any two diagrams of the same knot can be connected by a sequence of these moves.
Key insight: Reidemeister's Theorem states that two knot diagrams represent the same knot if and only if they can be connected by a finite sequence of twist, poke, and slide moves. This is the foundation of proving knot equivalence.
Search the catalog or filter by knots and links. In a name like 31, the main number counts crossings.
| Notation | Name | Crossings | Components | Chirality | Description |
|---|---|---|---|---|---|
| 0₁ | Unknot | 0 | 1 | achiral | The trivial knot - a simple closed loop. |
| 3₁ | Trefoil | 3 | 1 | chiral | The simplest nontrivial knot. Also called the overhand knot. |
| 4₁ | Figure-Eight | 4 | 1 | achiral | The unique 4-crossing knot. Symmetric and achiral. |
| 5₁ | Cinquefoil | 5 | 1 | chiral | A (5,2) torus knot with 5 crossings. |
| 5₂ | Three-Twist | 5 | 1 | chiral | The second simplest 5-crossing knot. |
| 6₁ | Stevedore | 6 | 1 | chiral | A 6-crossing twist knot named after a longshoreman's knot. |
| 6₂ | Miller Institute | 6 | 1 | chiral | A chiral 6-crossing knot. |
| 6₃ | Six-Three | 6 | 1 | achiral | The amphichiral 6-crossing knot — equivalent to its mirror image. |
| 7₁ | Septafoil | 7 | 1 | chiral | A (7,2) torus knot. |
| 2₁² | Hopf Link | 2 | 2 | chiral | Two circles linked together - the simplest link. |
| 4₁² | Solomon's Link | 4 | 2 | chiral | Two circles with 4 crossings — the (2,4) torus link, chiral. |
| 6₂³ | Borromean Rings | 6 | 3 | achiral | Three circles where no two are linked, but all three are inseparable. |
Browse a catalog of famous knots and links organized using Alexander-Briggs notation. The subscript indicates the crossing number and the superscript (if present) indicates the number of components.
Key insight: Knots are classified by their crossing number, chirality (whether mirror images are distinct), and whether they are alternating (crossings alternate over-under around the knot). Links are multi-component generalizations of knots.
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