The Tanglenomicon

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

Mathematics Department at The University of Iowa

Knots

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$\quad$
$\quad$

https://www.knotplot.com/

The natural question

How many knots?

Knot Tables

Lord Kelvin’s vortex theory of the atom

Atoms are knotted vortices in the æther.

By Hand

  • 1860’s Tait computes knots up to 7 crossings
    • 15 knots
  • 1870’s Tait, Kirkman, and Little compute knots up to 10 crossings
    • Takes about 25 years
    • 250 knots
  • 1960’s Conway computes knots up to 11 crossings
    • “A few hours”
    • 802 knots

By Computer

  • 1980’s Dowker and Thistlethwaite compute up to 13 crossings
    • First using a computer
    • 12,966 knots
  • 1990’s Hoste, Thistlethwaite, and Weeks compute up to 16 crossings
    • Computer runtime on the order of weeks
    • 1,701,936 knots
  • 2020’s Burton computes up to 19 crossings
    • 350 Million knots

Conway

How did Conway compute 25 years of work in "a few hours"?

Tangles

“We define a tangle as a portion of a knot diagram from which there emerge just 4 arcs pointing in the compass directions NW, NE, SW, SE.” - Conway, J.H.

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

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NWNESWSE
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Basic Operations

Operation $+$

$+$
$=$
$=$
$=$
$2$

Operation $\vee$

$\vee$
$=$
$=$
$=$
$\frac{1}{2}$

The Tanglenomicon

A table of two string tangles

(up to fixed boundary)

Building up

$\ $
$\ $
$\ $
$\ $

Where we are

Rational Tangles

8,388,608 up to 23 crossings

$\ $
$\begin{aligned}\to&\ \LP 3 \vee \frac{1}{2}\RP + 2\\&\\ \to&\ [3\ 2\ 2]\end{aligned}$

Montesinos

120,344,744 up to 23 crossings

with non-fixed boundary

$+$
$=$
$$\ =\ $$
$$=[3\ 2\ 0] + [3\ 2\ 0]$$

Generalized Montesinos

Operation $\circ$

$\ $
$= \color{var(--r-Purple)}([1\ 2\ 0] + [1\ 2\ 0] + [1\ 1\ 0]) \color{var(--r-Foreground)}\circ \color{var(--r-Red)}[2\ 2]$

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.

Where we’re going

Algebraic (Arborescent)

All possible tangles made from $+$ and $\vee$ on basic tangles

Algebraic
A tangle build from $\vee$ and $+$ on some rational tangles.
$$\LP\color{var(--r-Purple)}\LB3\ 2\ 3\RB+\LB3\ 2\ 3\RB\color{var(--r-Foreground)}\RP\vee\LP\color{var(--r-Purple)}\LB3\ 2\ 3\RB+\LB3\ 2\ 3\RB\color{var(--r-Foreground)}\RP$$

Arborescent Tangles are constructed by taking a collection of twisted bands described by a weighted tree and connecting them with successive plumbing.

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

3 2 -3 0 4 3

Into the future

Non-algebraic/Polygonal

4-valent planar graphs

$\quad$

4-valent planar graph insertions

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

Technologies

ThrowTheSwitch/UnitySimple Unit Testing for C C3.3k935

joe-starr.com

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
  16. Connolly, Nicholas. Classification and Tabulation of 2-String Tangles: The Astronomy of Subtangle Decompositions. University of Iowa, 2021, https://doi.org/10.17077/etd.005978.

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