The Tanglenomicon

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

Mathematics Department at The University of Iowa

Knots

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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

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Where we are

Rational Tangles

8,388,608 up to 23 crossings

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

Montesinos

120,344,744 up to 23 crossings

with non-fixed boundary

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

Generalized Montesinos