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

$$[3\ 2\ 2]$$

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

Montesinos Tangles

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

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

Generalized Montesinos Tangles

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

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

Not Classified

Algebraic Tangles

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

Non-Algebraic Tangles 

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

Sources

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