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The Tanglenomicon

A table of two string tangles

The algebraic tangles

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

Mathematics Department at The University of Iowa

Joint Math Meetings 2025 (1/10/25)

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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Basic Operations

Operation $+$

$+$

$=$

$2$

Operation $\vee$

$\vee$

$=$

$\frac{1}{2}$

Algebraic Tangles

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

Algebraic

A tangle build from $\vee$ and $+$ on some rational tangles.

$$\LP\color{var(--r-Purple)}\LB3\ 2\ 3\RB+\LB3\ 2\ 3\RB\color{var(--r-Foreground)}\RP\vee\LP\color{var(--r-Purple)}\LB3\ 2\ 3\RB+\LB3\ 2\ 3\RB\color{var(--r-Foreground)}\RP$$

But actually arborescent

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

It’s straight forward to see (you should see in the example) that algebraic and arborescent constructions describe the same class of object.

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

Anatomy of a tree

Rings

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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

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Definition

The count of rings attached to a vertex is the Ring Number of the vertex. Ring numbers are noted as an augmentation of the vertex.

Definition

A vertex with ring number $\geq 1$ or valence $\geq 3$ is called an Essential Vertex.

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

Definition

The subtrees remaining after excising all essential vertices and their bonds (half edges) are called the Sticks of a tree.

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

Linearization of Arborescent tangles

$\pm$ Abbreviated Canonical Tangle Trees

Weight Condition At each vertex of $\Gamma$, at most one weight is non-zero.
Stick Condition On any stick the weights are non-zero except for end vertices that have a bond free in $\Gamma_0$ and for the case $\Gamma_0=\alpha(0[0])$, $\Gamma_0=\alpha[0]$. The non-zero weights along any stick are of alternating sign. No end vertex of a stick has weight $\pm 1$ unless it has a bond free in $\Gamma_0$, or $\Gamma_0=[\pm 1]$.
One of:
- Positivity Condition There are no sticks in $\Gamma_0$ of the form $[-1],\ [ -2],\ \alpha[-2],\ \alpha[2]\alpha$.
- Negative Condition There are no sticks in $\Gamma_0$ of the form $[1],\ [ 2],\ \alpha[2],\ \alpha[2]\alpha$
Abbreviation Condition A vertex of $\Gamma_0$ of ring number $1$ has valence $\geq 2$, and $\Gamma_0$ is not $\alpha\langle 2\ 0\rangle$. Every ring subtree of $\Gamma$ is adjacent to a non-essential vertex.

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

Theorem (Bonahon and Siebenmann)

Consider two $\pm$ canonical abbreviated arborescent tangle trees $\Gamma$ and $\Gamma^\prime$. Plumbing according to $\Gamma$ and $\Gamma^\prime$ gives isomorphic arborescent tangles if and only if $\Gamma$ and $\Gamma^\prime$ can be deduced from each other by a sequence of moves in the calculus of arborescent tangle trees (happy to talk about the calculus another time).

Note

Note $\pm$ abbreviated canonical tangle trees do not describe minimal diagrams.

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

Definition

We call the crossing number of a the tangle diagram described by a $\pm$ abbreviated canonical tangle tree the canonical arborescent crossing number (CACN) which is given by: $$\text{CACN}=4\cdot\#(\text{rings in the tree})+\sum |\text{weights}|$$

Rational Tangle Trees

Montesinos Tangle Trees

Constructing an arborescent tangle

Game plan

Find a way to generate abstract rooted trees
Modify that method to get tangle trees

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Nakano, S. (2002). Efficient generation of plane trees. In Information Processing Letters (Vol. 84, Issue 3, pp. 167-172). Elsevier BV. https://doi.org/10.1016/s0020-0190(02)00240-5

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