Language: English Updated: 2026-06-01

List of knots

A list of knots is a compiled catalogue that enumerates known knots—closed, non-self-intersecting curves in three‑dimensional space—along with relevant classifications, properties, and references. Such lists serve as reference tools for mathematicians, physicists, chemists, and practitioners in fields such as ropework, sailing, climbing, and decorative arts.

Purpose and Scope
The primary aim of a list of knots is to organize the extensive variety of knot types discovered through mathematical study, practical application, or historical documentation. It typically includes:

  • Mathematical knots: Idealized, smooth embeddings of a circle $S^1$ in $\mathbb{R}^3$ considered up to ambient isotopy.
  • Prime and composite knots: Classification into knots that cannot be expressed as a non‑trivial connected sum (prime) and those that can (composite).
  • Knot invariants: Information such as crossing number, Alexander polynomial, Jones polynomial, knot group, and hyperbolic volume.
  • Notation systems: Conventional identifiers like the Alexander–Briggs notation (e.g., $3_1$ for the trefoil) and the Dowker–Thistlethwaite code.
  • Practical knots: Commonly used rope or cordage knots (e.g., bowline, figure‑eight, clove hitch) with descriptions of tying methods, typical uses, and load‑bearing characteristics.

Typical Structure

  1. Introduction – Brief overview of knot theory and practical knotwork.
  2. Classification by Crossing Number – Tables listing all prime knots up to a given crossing number (often up to 10 or 12), with accompanying invariants.
  3. Special Families – Sections on torus knots, satellite knots, alternating knots, and hyperbolic knots.
  4. Composite Knots – Descriptions of knots formed by the connected sum of prime knots.
  5. Practical Knot Catalogues – Organized by function (e.g., binding, loop, hitch) with step‑by‑step tying instructions and safety notes.
  6. References and Further Reading – Citations to standard texts such as The Knot Book by Colin C. Adams, Knots and Links by Dale Rolfsen, and databases like the Knot Atlas and the Hoste–Thistlethwaite table.

Major Sources and Databases

  • The Knot Atlas – An online repository providing diagrams, invariants, and downloadable data for thousands of knots.
  • KnotInfo – A curated table of knot invariants maintained by the University of Texas at Austin.
  • Rolfsen’s Table – The classic enumeration of all prime knots through ten crossings, published in 1976.
  • The International Association of Knot Tying (IAKT) – Provides standardized names and tying instructions for practical knots used in climbing, sailing, and rescue.

Historical Development

The systematic enumeration of mathematical knots began in the late 19th century with the work of Peter Guthrie Tait, who catalogued knots up to ten crossings using hand‑drawn projections. Subsequent advances in topology, notably the development of knot invariants in the 20th century (e.g., Alexander polynomial, Jones polynomial), enabled more refined classifications. In parallel, practical knot compendia emerged from seafaring and mountaineering traditions, leading to modern, cross‑disciplinary listings that integrate both mathematical and utilitarian perspectives.

Applications

  • Mathematics and Physics – Knot tables assist in research on low‑dimensional topology, quantum field theory, and polymer science.
  • Biology – Classification of DNA and protein entanglements relies on knot inventories.
  • Engineering and Safety – Proper selection of binding, loop, and hitch knots is critical in climbing, rescue operations, and rigging.

Limitations

While extensive, any list of knots is inherently incomplete due to ongoing discovery of new knot types, particularly in the realm of high‑crossing-number mathematical knots and novel practical knots adapted for emerging technologies.

See Also

  • Knot theory
  • Prime knot
  • Knot invariants
  • Practical knot tying
  • List of rope knots

References

  1. Adams, C. C. (1994). The Knot Book. American Mathematical Society.
  2. Rolfsen, D. (1976). Knots and Links. Publish or Perish, Inc.
  3. Hoste, J., Thistlethwaite, M., & Weeks, J. (1998). "The first 1,701,936 knots". Mathematical Proceedings of the Cambridge Philosophical Society, 124(1), 27–39.
  4. The Knot Atlas. https://katlas.org
  5. KnotInfo. https://knotinfo.math.indiana.edu

This entry provides a concise, factual overview of the concept of a list of knots as it is recognized in both mathematical and practical contexts.

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