`$\newcommand{\N}{\mathbb{N}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\R}{\mathbb{R}} \newcommand{\LP}{\left(} \newcommand{\RP}{\right)} \newcommand{\LS}{\left\lbrace} \newcommand{\RS}{\right\rbrace} \newcommand{\LA}{\left\langle} \newcommand{\RA}{\right\rangle} \newcommand{\LB}{\left[} \newcommand{\RB}{\right]} \newcommand{\MM}{\ \middle|\ } \newcommand{\abs}[1]{\left\vert#1\right\vert} \newcommand{\msr}[1]{m\left(#1\right)} \newcommand{\inv}[1]{#1^{-1}} \newcommand{\bkt}[1]{\LA \img{#1}\RA} \require{color}$`

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

$\circ$

Rational Tangles

[3 2 2]

$$\to$$

$+$

Montesinos Tangles

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

$$\to$$

$\circ$

Generalized Montesinos Tangles

([3 0] + [3 0] + [2 0]) $\circ$ (1,2)

Not Classified

$+\ \vee$

Algebraic Tangles

([3 2 3] + [3 2 3]) ∨ ([3 2 3] + [3 2 3] )

$$\to$$

Non-Algebraic Tangles

6* *.[3 2 3 1].[3 2 3 1].[3 2 3 1].[3 2 3 1].[3 2 3 1]

Architecture

Sources