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The Knot Zoo is a web gallery of mathematical knots from the KnotPlot project. Select a displayed knot and the site loads it into an interactive 3D viewer. Unlike an ordinary piece of string, a mathematical knot is a closed loop with its ends joined, studied up to continuous deformation rather than as a tied object with free ends.
What The Knot Zoo is
KnotPlot, created by Robert G. Scharein, presents knots and links from a mathematical and visual perspective. “A Knot Zoo” is the gallery interface: it lets you browse examples and inspect their three-dimensional forms. It is not a physical zoo, a rope-tying kit or a single manufactured product.
The gallery page instructs visitors to click a knot to load it into an interactive 3D viewer and states that JavaScript must be enabled. The page records a last-modified date of 9 December 2020; current browser behavior is not independently established here.
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How to explore a knot
- Open A Knot Zoo in a browser with JavaScript enabled.
- Choose one of the displayed knot entries.
- Click the knot to load it in the interactive 3D viewer.
- Rotate the model to inspect the three-dimensional path and compare it with other entries.
The gallery is primarily a visual browsing tool. Its purpose is to make abstract knot classifications easier to see, not to teach physical knot-tying techniques.
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How knots are ordered
Crossing number
KnotPlot describes its catalogue as arranged by increasing complexity, using crossing number as one common measure. A knot’s crossing number is the smallest number of double points that appear in any planar projection of that knot. A drawing can show more crossings than the knot’s crossing number if the projection is not minimal.
Crossing number is useful for ordering examples, but it is not a complete definition of complexity. Different knots can share the same crossing number, and other mathematical measurements may distinguish them.
Knots and links
A knot has one closed component. A link has two or more closed components that may be interlaced. KnotPlot uses both terms, so a catalogue or software count that says “knots and links” should not be read as a count of knots alone.
Published catalogue figures
KnotPlot’s mathematical-knots pages report the following figures. They have different scopes and are not a single combined total.
| Statement | What it covers | Date qualification |
|---|---|---|
| 12,965 knots with 13 or fewer crossings in a minimal projection | Knots meeting the stated crossing threshold | Year not stated on the page |
| 1,701,935 knots with 16 or fewer crossings in a minimal projection | Knots meeting the higher crossing threshold | Year not stated on the page |
| More than 3,000 knots and links in the KnotPlot software database | The software database, including links | Year not stated on the page |
These numbers are statements published by KnotPlot, not newly verified counts. The Zoo interface may expose only part of the broader catalogue, and the available pages do not establish whether the figures have been updated since publication.
Gallery, diagram and software: what differs?
| Way to explore | What you get | Best for |
|---|---|---|
| Mathematical classification | Ordering and comparison using measures such as crossing number | Understanding how examples are catalogued |
| Static knot diagram | A planar drawing that can hide the full three-dimensional shape | Reading crossings and studying notation |
| A Knot Zoo browser gallery | Clickable entries that open an interactive 3D viewer | Visual browsing without installing software |
| KnotPlot software | Tools for visualizing and manipulating mathematical knots in three and four dimensions; its database is described as containing more than 3,000 knots and links | Hands-on exploration and manipulation |
A rotatable model is not a replacement for a mathematical diagram: projection conventions and crossing information still matter. Conversely, a flat diagram cannot always convey how the closed curve occupies three-dimensional space.
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Using the KnotPlot catalogue command
The KnotPlot manual documents a software command, zoo start size, for displaying a size-by-size selection beginning at an entry in the standard catalogue. If size is omitted, it defaults to five; a start value of zero selects randomly. This documentation describes the software command, not necessarily the controls of the current web gallery.
What The Knot Zoo is not
- It is not evidence of a physical Amazon product or a boxed educational kit.
- It is not a catalogue of ordinary open-ended string knots.
- It is not proof that every knot in KnotPlot’s wider database appears in the browser gallery.
- It is not a usability benchmark against rival services; the available material does not establish such a comparison.
Why the gallery is useful
For beginners, the Zoo provides an immediate visual link between an abstract name, a planar classification and a three-dimensional object. For students, it offers examples to compare as crossing number increases. For experienced users, it serves as a quick visual index into KnotPlot’s broader modelling environment.
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