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🪢Knot Theory

9.1 Historical development of knot tables

3 min readLast Updated on July 22, 2024

Knot tables have come a long way since Tait's first attempt in 1877. From manual tabulation to computerized databases, these tables have grown to include knots with up to 19 crossings, revolutionizing how we study and classify knots.

The development of knot tables has faced challenges like distinguishing between knots and computational limitations. However, these tables have become crucial for classification, developing invariants, and applications in physics and chemistry, connecting knot theory to various mathematical disciplines.

Historical Development of Knot Tables

Progression of knot tabulation

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  • Early attempts at knot tabulation
    • Tait's knot tables (1877) first systematic attempt to classify knots included knots up to 10 crossings
    • Little's knot tables (1885) extended Tait's work to include knots up to 11 crossings
  • Modern comprehensive knot tables
    • Rolfsen's knot table (1976) included knots up to 10 crossings with detailed diagrams and invariants
    • Hoste-Thistlethwaite-Weeks (HTW) knot table (1998) computerized table of prime knots up to 16 crossings
    • Knot Atlas (2005-present) online database of knots and links includes knots up to 19 crossings and links up to 12 crossings

Contributors to knot tabulation

  • Peter Guthrie Tait (1831-1901)
    • Scottish physicist and mathematician pioneered the systematic study of knots
    • Created the first knot tables up to 10 crossings
  • Charles Newton Little (1858-1923)
    • American mathematician extended Tait's work to include knots up to 11 crossings
    • Introduced the concept of knot invariants
  • Dale Rolfsen (1940-present)
    • Canadian mathematician created a comprehensive knot table with detailed diagrams and invariants
    • Author of the influential book "Knots and Links"
  • Jim Hoste, Morwen Thistlethwaite, and Jeff Weeks collaborated to create the computerized HTW knot table extended the tabulation of prime knots up to 16 crossings
  • Dror Bar-Natan and Scott Morrison creators of the online Knot Atlas database continuously update and maintain the database with contributions from the knot theory community

Challenges in early tabulation

  • Distinguishing between different knots
    • Early tabulators relied on visual inspection and manual manipulation of knot diagrams
    • Development of knot invariants (Alexander polynomial, Jones polynomial) provided more reliable methods for distinguishing knots
  • Computational limitations
    • Manual tabulation was time-consuming and error-prone
    • Advances in computer technology and algorithms allowed for automated generation and analysis of knot diagrams
  • Incomplete or inconsistent notation
    • Early knot tables lacked a standardized notation system
    • Introduction of Dowker-Thistlethwaite notation and Gauss code provided a consistent way to represent knots
  • Limited scope
    • Early tables only included knots up to a certain number of crossings
    • Collaborative efforts and use of computers enabled tabulation of knots with higher crossing numbers

Significance of knot tables

  • Classification and organization of knots
    • Knot tables provide a systematic way to categorize and study knots
    • Enable researchers to identify patterns and relationships between different knots
  • Development of knot invariants
    • Knot tables serve as a testing ground for new knot invariants
    • Help in evaluating the effectiveness and discriminatory power of invariants
  • Applications in physics
    • Knot tables are used to study the behavior of knotted structures in polymer physics and statistical mechanics
    • Help in understanding the properties of DNA and other biological molecules
  • Applications in chemistry
    • Knot tables are used to classify and study molecular knots and links
    • Aid in the design and synthesis of novel knotted molecules with potential applications in materials science and drug delivery
  • Connections to other areas of mathematics
    • Knot tables provide insights into the relationships between knot theory and other mathematical disciplines (group theory, topology, combinatorics)
    • Stimulate cross-disciplinary research and the development of new mathematical tools and techniques


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© 2025 Fiveable Inc. All rights reserved.
AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website.