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GEOTOP-A International Conference (1/11/24)

The Tanglenomicon

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

Mathematics Department at The University of Iowa

Knot Tables

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

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

KnotInfo

Conway

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.”

Basic Operations

Operation $+$

Operation $\vee$

The Tanglenomicon

The table of two string tangles

Building up

Where we are

Rational Tangles

Generation

For any $N$ an obvious twist vector is the twist vector of all $1$s $$[1\ 1\ 1\ \cdots\ 1]$$ Noting that when we write this sequence, we have $N-1$ spaces.

If we choose to place a $+$ instead of the left most space we get $$[1+1\ 1\ \cdots\ 1]=[2\ 1\ \cdots\ 1]$$ we’re free to make this choice for each space

this gives $N-1$ choices between ‘$+$’ and space $$[1\square 1\square 1\square\cdots\square1]$$ letting us generate twist vectors by simply counting from $0\to 2^{N-1}$.

Canonical Twist Vectors

We can write a canonical twist vector by taking the odd length vectors (appending $0$ where needed).

Computations

Rational Number (continued fraction)

The rational number for a twist vector is computed by taking the twist vector as a finite continued fraction that is: $$\LB a\ b\ c\RB=c+\frac{1}{b+\frac{1}{a}}$$

Using The Tanglenomicon

Where we’re going

Montesinos

Existence of canonical diagrams for Montesinos tangles

Generation

The Montesinos tangles of crossing number $N$ have a slightly simpler generation strategy compared to rational tangles. We again generate twist vectors but require that each entry $e$ of the twist vector satisfies $2\leq e < N.$ We call these restricted set of twist vectors stencils.

Now for each entry $e_i$ of the stencil, we generate a list of rational tangles of crossing number equal to $e_i$, with the restriction $0<\frac{p_i}{q_i}<1$. We then take all combinations of elements of these lists.

What about the ‘k’?

The construction for the canonical Montesinos tangles includes a trailing $\frac{k}{1}$ tangle. Our generation strategy seems to miss these.

What we’re actually generating with this algorithm is equivalent to allowing the boundary components of the tangle to move. To recover fixed boundary tangles we need to generate the next larger class of tangles.

We can note which stage a tangle was generated at to allow users to choose datasets for fixed or non-fixed boundary tangles.

Generalized Montesinos

Operation $\circ$

Generation

We just need to take our lists of Montesinos and rational tangles and glue them together with $\circ$.

Into the future

Algebraic

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

Generation

Algebraic Tangle Trees

To generate all possible algebraic tangles, we can generate all possible algebraic expressions on the trivial tangles. Equivalently, all full binary trees with $N$ leaves. Where the tree’s internal nodes are labeled with combinations of $\vee$ and $+$ and leaves are labeled with all combinations of trivial tangles.

These binary trees are called Algebraic Tangle Trees.

Non-algebraic/Polygonal

4-valent planar graphs

4-valent planar graph insertions

Generation

There exist tables of 4 valent graphs. We can use those with insertions from our list of algebraic tangles to generate all polygonal tangles.

Technologies

Sources

Programmatic Description

Computations

Rational Number (continued fraction)

The rational number for a twist vector is computed by taking the twist vector as a finite continued fraction that is: $$\LB a\ b\ c\RB=c+\frac{1}{b+\frac{1}{a}}$$