The Tanglenomicon

Zachary Bryhtan, Nicholas Connolly, Isabel Darcy, Ethan Rooke, Joseph Starr*

Mathematics Department at The University of Iowa

Tangle tabulation

“The Most Important Missing Infrastructure Project in Knot Theory” -Dr. Dror Bar-Natan [2012]

Classified

Rational Tangles

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$$[3\ 2\ 2]$$

$$([1\ 2\ 0] + [1\ 2\ 0] + [1\ 1\ 0]) \circ (2,2)$$
$$\rightarrow$$

Montesinos Tangles

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$$[3 0 ] + [2 1 0] + [2 2 0]$$

$$([1\ 2\ 0] + [1\ 2\ 0] + [1\ 1\ 0]) \circ (2,2)$$
$$\rightarrow$$

Generalized Montesinos Tangles

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$$([1\ 2\ 0] + [1\ 2\ 0] + [1\ 1\ 0]) \circ (2,2)$$

$$([1\ 2\ 0] + [1\ 2\ 0] + [1\ 1\ 0]) \circ (2,2)$$

Not Classified

Algebraic Tangles

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$$([3\ 2\ 3] + [3\ 2\ 3]) \vee ([3\ 2\ 3] + [3\ 2\ 3] )$$
$$6^\ast\ *.[3\ 2\ 3\ 1].[3\ 2\ 3\ 1].[3\ 2\ 3\ 1].[3\ 2\ 3\ 1].[3\ 2\ 3\ 1]$$
$$\rightarrow$$

Non-Algebraic Tangles

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$$6^\ast\ *.[3\ 2\ 3\ 1].[3\ 2\ 3\ 1].[3\ 2\ 3\ 1].[3\ 2\ 3\ 1].[3\ 2\ 3\ 1]$$
$$6^\ast\ *.[3\ 2\ 3\ 1].[3\ 2\ 3\ 1].[3\ 2\ 3\ 1].[3\ 2\ 3\ 1].[3\ 2\ 3\ 1]$$

Sources

  1. Dror Bar-Natan The Most Important Missing Infrastructure Project in Knot Theory
  2. Kauffman, L. H., and S. Lambropoulou. “From Tangle Fractions to DNA.” In Topology in Molecular Biology, edited by Michail Ilych Monastyrsky, 69-110. Biological and Medical Physics, Biomedical Engineering. Berlin, Heidelberg: Springer Berlin Heidelberg, 2007. https://doi.org/10.1007/978-3-540-49858-2_5.
  3. Moon, Hyeyoung, and Isabel K. Darcy. “Tangle Equations Involving Montesinos Links.” Journal of Knot Theory and Its Ramifications 30, no. 08 (July 2021): 2150060. https://doi.org/10.1142/S0218216521500607.
  4. Conway, J.H. “An Enumeration of Knots and Links, and Some of Their Algebraic Properties.” In Computational Problems in Abstract Algebra, 329-58. Elsevier, 1970. https://doi.org/10.1016/B978-0-08-012975-4.50034-5.
  5. Louis H. Kauffman and Sofia Lambropoulou. Classifying and applying rational knots and rational tangles. In DeTurck, editor, Contemporary Mathematics, volume 304, pages 223-259, 2001
  6. Alain Caudron. Classification des nœuds et des enlacements, volume 4 of Publications Math ́ematiques d’Orsay 82 [Mathematical Publications of Orsay 82]. Universit ́e de ParisSud, D ́epartement de Mathe ́matique, Orsay, 1982.
  7. Robert Glenn Scharein. Interactive topological drawing. ProQuest LLC, Ann Arbor, MI, 1998. Thesis (Ph.D. The University of British Columbia (Canada). URL: https://www.knotplot.com/.
  8. Jablan, S., & Sazdanović, R. (2007). Linknot. In Series on Knots and Everything. WORLD SCIENTIFIC. https://doi.org/10.1142/6623
  9. Dowker, C. H., & Thistlethwaite, M. B. (1983). Classification of knot projections. In Topology and its Applications (Vol. 16, Issue 1, pp. 19-31). Elsevier BV. https://doi.org/10.1016/0166-8641(83)90004-4
  10. Hoste, J., Thistlethwaite, M., & Weeks, J. (1998). The first 1,701,936 knots. In The Mathematical Intelligencer (Vol. 20, Issue 4, pp. 33-48). Springer Science and Business Media LLC. https://doi.org/10.1007/bf03025227
  11. Burton, B. A. (2020). The Next 350 Million Knots. Schloss Dagstuhl - Leibniz-Zentrum Für Informatik. https://doi.org/10.4230/LIPICS.SOCG.2020.25
  12. C. Livingston and A. H. Moore, KnotInfo: Table of Knot Invariants, knotinfo.math.indiana.edu, today’s date (eg. August 24, 2023).
  13. Schubert, Horst. “Knoten mit zwei Brücken..” Mathematische Zeitschrift 65 (1956): 133-170. http://eudml.org/doc/169591.
  14. Jos ́e M. Montesinos. Seifert manifolds that are ramified two-sheeted cyclic coverings. Bol. Soc. Mat. Mexicana (2), 18:1-32, 1973.
  15. F. Bonahon and L. Siebenmann, New geometric splittings of classical knots, and the classification and symmetries of arborescent knots, http://www-bcf.usc.edu/~fbonahon/Research/Publications.html