Knots, links, and the fundamental Reidemeister moves
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.
The simplest non-trivial knot with 3 crossings.
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's fundamentally different from the unknot!
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.
The simplest non-trivial knot with 3 crossings.
The crossing number is the minimum number of crossings in any diagram of the knot. It's a topological invariant - no matter how you deform the knot, you can't reduce the number of crossings below this minimum! The unknot has crossing number 0, the trefoil has crossing number 3.
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!
You can add or remove a simple twist without changing the knot.
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 these three moves (plus planar isotopy). This is the foundation of proving knot equivalence!
Browse a catalog of famous knots and links. Knots are organized using Alexander-Briggs notation, where the subscript indicates the crossing number and the superscript (if present) indicates the number of components.
| Notation | Name | Crossings | Components | Chirality | Description |
|---|---|---|---|---|---|
| 0ā | Unknot | 0 | 1 | achiral | The trivial knot - a simple closed loop. |
| 3ā | Trefoil | 3 | 1 | chiral | The simplest nontrivial knot. Also called the overhand knot. |
| 4ā | Figure-Eight | 4 | 1 | achiral | The unique 4-crossing knot. Symmetric and achiral. |
| 5ā | Cinquefoil | 5 | 1 | chiral | A (5,2) torus knot with 5 crossings. |
| 5ā | Three-Twist | 5 | 1 | chiral | The second simplest 5-crossing knot. |
| 6ā | Stevedore | 6 | 1 | achiral | A symmetric 6-crossing knot used by longshoremen. |
| 6ā | Miller Institute | 6 | 1 | achiral | An achiral 6-crossing knot. |
| 6ā | Six-Three | 6 | 1 | chiral | A chiral 6-crossing knot. |
| 7ā | Septafoil | 7 | 1 | chiral | A (7,2) torus knot. |
| 2āĀ² | Hopf Link | 2 | 2 | chiral | Two circles linked together - the simplest link. |
| 4āĀ² | Solomon's Seal | 4 | 2 | achiral | Two circles with 4 crossings, achiral. |
| 6āĀ³ | Borromean Rings | 6 | 3 | achiral | Three circles where no two are linked, but all three are inseparable. |
Chiral: The knot has a distinct left-handed and right-handed version (mirror images are different).
Achiral: The knot is identical to its mirror image.
nā: The first knot with n crossings
nāĀ²: The first link with n crossings and 2 components
1. 3D Visualization: We explored famous knots in 3D - the unknot, trefoil, figure-eight, cinquefoil, and Hopf link. Each has a unique 3D structure that can be continuously deformed.
2. Knot Diagrams: We learned how to represent knots as 2D projections with over/under crossing information. The crossing number is a fundamental invariant.
3. Reidemeister Moves: We discovered the three fundamental operations (twist, poke, slide) that preserve knot equivalence. These are the building blocks of proving two knots are the same!
4. Knot Catalog: We browsed a table of famous knots using Alexander-Briggs notation. We learned about chirality, alternating knots, and the distinction between knots and links.
Next Up: Now that you understand the basics of knots, you're ready to explore knot invariants - mathematical properties that help us distinguish knots. You'll learn about tricolorability, the unknotting number, and how to prove that the trefoil is truly knotted!