Defense Slides (8/18/25)

The Tanglenomicon:

Tabulation of Arborescent Tangles

Joseph Starr

Mathematics Department at The University of Iowa

Partially supported by DMS-2038103

"The Tanglenomicon" name due to Dr. Nicholas Connolly

Knots

$\quad$

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

1860s Tait computes knots up to 7 crossings
- 15 knots
1870s Tait, Kirkman, and Little compute knots up to 10 crossings
- Takes about 25 years
- 250 knots (+1 repeat in the Perko pair)
1960s Conway computes knots up to 11 crossings
- “A few hours”
1980s Caudron verifies knots up to 11 crossings
- Finding 4 omissions of Conway

By Computer

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

$\quad$

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 built from $\vee$ and $+$ on some basic tangles.

$$\color{var(--r-foreground)}\LP\color{var(--r-Purple)} \LP\LP3\vee\\\frac{1}{2}\RP+3\RP+\LP\LP3\vee\frac{1}{2}\RP+3\RP\color{var(--r-foreground)}\RP \vee\color{var(--r-foreground)}\LP\color{var(--r-Purple)} \LP\LP3\vee\frac{1}{2}\RP+3\RP+\LP\LP3\vee\frac{1}{2}\RP+3\RP\color{var(--r-foreground)}\RP $$ $$\vee \color{var(--r-Purple)}++\vee3\frac{1}{2}3+\vee3\frac{1}{2}3 ++\vee3\frac{1}{2}3+\vee3\frac{1}{2}3 $$

Arborescent Tangles

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

Algebraic and arborescent constructions describe the same class of objects.

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

$\ $

We can see here the correspondence between algebraic and arborescent constructions.

How to encode? (Bonahon and Siebenmann)

Relationship between a band and the other bands plumbed to it (children).
Location of twists relative to the children.
Tangle boundary.

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

1. Relationship between a band and the other bands attached to it (children)

Acyclic connections between items.
Relative positions of connections.

Acyclic connections between items?

Solution: A tree (in the graph sense)

Relative positions of connections?

Solution: Rooted plane tree

Definition

A rooted plane tree is an abstract tree imbued with a strict total order on the vertices (indexed by the non-negative integers). We call the least vertex the root of the tree.

Convention:

We will select the total order given by a depth first in order traversal of the tree.

Cyclic Order of a Vertex

$\quad$

How to encode? (Bonahon and Siebenmann)

✓ Relationship between a band and the other bands attached to it (children).
Location of twists relative to the children.
Tangle boundary.

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

$\quad$

How to encode? (Bonahon and Siebenmann)

✓ Relationship between a band and the other bands attached to it (children).
✓ Location of twists relative to the children.
Tangle boundary.

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

Local view of a vertex

Bonds

Free Bonds

$\ $

Rotations

Elements of Klein four-group ($V_4$)

$\ $

How to encode? (Bonahon and Siebenmann)

✓ Relationship between a band and the other bands attached to it (children).
✓ Location of twists relative to the children.
✓ Tangle boundary.

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

Weighted planar tangle tree

Anatomy of a Weighted planar tangle tree

Rings

$\ $

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

A vertex with 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

Moves on a Weighted Planar Tangle Tree

Definition: Part 1

The $F_3^\prime$ move on a weighted arborescent tree moves a weight $w$ as in the image below and, if $w$ is odd, reverse the cyclic order of weights and bonds at all vertices of the purple subtree lying at odd distance (count of edges between two vertices) from the vertex shown.

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

The flype

“if $w$ is odd, reverse the cyclic order of weights and bonds at all vertices of the purple subtree lying at odd distance (count of edges between two vertices) from the vertex shown.”

Definition: Part 2

Also, when $w$ is odd, apply $\xi$ ( $X$-axis rotation) to all free bonds in the purple subtree that are attached to a vertex at even distance from the vertex shown, and $\eta$ ($Y$-axis rotation) to those at odd distance. The rotations are relative to the local orientations of the plumbing squares on the bands corresponding to vertices at even/odd distances from the vertex with weight $w$.

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 $F_2$ move on a weighted arborescent tangle tree reverses the cyclic order of bonds and weights at one vertex on the tree and at every vertex at even distance from it; also apply $\eta$ ($Y$-axis rotation) to every free bond of a vertex at even (or zero) distance, and apply $\xi$ ($X$-axis rotation) to every free bond at odd distance.

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 $F_1$ move on a weighted arborescent tangle tree reverses the cyclic order of bonds and weights at every vertex of the graph and applies $\zeta$ ($Z$-axis rotation) to every free bond.

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 $R^-$ replaces the left image below with the right, leaving the rest of the tree unchanged.

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 $R^+$ replaces the left image below with the right, leaving the rest of the tree unchanged.

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

Canonical Trees

For our purposes, a weighted planar tree $\Gamma$ is called a canonical weighted planar tangle tree (CWPTT) if it has a single free bond with a label from $V_4$ and satisfies the following conditions.

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

Weight Condition

At each vertex of $\Gamma$, at most one weight is non-zero.

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

Stick Conditions

On any stick the weights of the vertices are non-zero except for end vertices that have a bond free in $\Gamma$ and for the case $\Gamma$ is:

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 $\Gamma$.

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

One of

Positivity Condition $\LP+\RP$-CWPTT

Except for those with a free bond, there are no sticks in $\Gamma$ of the forms

or

Negativity Condition $\LP-\RP$-CWPTT

Except for those with a free bond, there are no sticks in $\Gamma$ of the forms

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 Seibenmann

There exists an effective algorithm which, for any weighted planar tree $\Gamma$ with free bonds labeled by elements of $V_4$, alters $\Gamma$ by a sequence of moves of the calculus of arborescent tangles to produce a collection of positively (or negatively) canonical weighted planar trees.

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 Seibenmann

Consider two positive (or negative) CWPTT $\Gamma$ and $\Gamma^\prime$, with free bonds labeled by elements of $V_4$. 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 ($F_1$), ($F_2$), ($F_3^\prime$), and the modified ring moves $\LP\pm R\RP$.

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

For weighted planar tangle tree $\Gamma$, with weights $\LS w_i\RS$. We call $$\text{TCN}=\sum |w_i|$$ the Tree Crossing Number (TCN).

CWPTT are Not Minimal

$$\to$$

A Canonical Vertex [S]

A vertex $v_i$ of a weighted planar tangle tree $\Gamma$ with a single free bond labeled from $V_4$ is said to be $\LP+\RP$-canonical if $v_i$ has at most one non-zero weight $w_i$ and $i$ is:

Zero ($v_0$ the root) with no further conditions.
Non-zero with the following conditions satisfied:

I. If the valence of $v_i$ is $1$ then all of:

(Stick Condition)
1. $w_i\neq 0$ unless $i=1$ and $w_0=0$ (the weight of the root)
2. $w_i\neq \pm 1$
3. If the valence of $v_{i-1}$ (the parent) is $2$ then $\text{sign}\LP w_i\RP\neq\text{sign}\LP w_{i-1}\RP$ unless $i=1$ and $w_0=0$
(Positivity Condition)
1. If the valence of $v_{i-1}$ (the parent) is greater than $2$ then $w_i\neq -2$

Stick Condition
1. On any stick the weights of the vertices are non-zero except for end vertices that have a bond free in $\Gamma$ and for the case $\Gamma$ is:

$$\text{or}$$

2. The non-zero weights along any stick are of alternating sign.
3. No end vertex of a stick has weight $\pm 1$ unless it has a bond free in $\Gamma$.

I. If the valence of $v_i$ is $1$ then all of:

(Stick Condition)
1. $w_i\neq 0$ unless $i=1$ and $w_0=0$ (the weight of the root)
2. $w_i\neq \pm 1$
3. If the valence of $v_{i-1}$ (the parent) is $2$ then $\text{sign}\LP w_i\RP\neq\text{sign}\LP w_{i-1}\RP$ unless $i=1$ and $w_0=0$

Positivity Condition:
Except for those with a free bond, there are no sticks in $\Gamma$ of the forms

$$\text{or}$$

I. If the valence of $v_i$ is $1$ then all of:

(Positivity Condition)
1. If the valence of $v_{i-1}$ (the parent) is greater than $2$ then $w_i\neq -2$

II. If the valence of $v_i$ is $2$ then all of:

(Stick Condition)
1. $w_i\neq 0$
2. If valence of the parent or valence of the child is greater than $2$ (is essential) then $w_i\neq \pm1$
3. $\ $
  1. If valence of the parent is $2$ then $\text{sign}\LP w_i\RP\neq\text{sign}\LP w_{i-1}\RP$ (the parent)
  2. If valence of the child is $2$ then $\text{sign}\LP w_i\RP\neq\text{sign}\LP w_{i+1}\RP$ (the child)
(Positivity Condition)
1. If valence of the parent and valence of the child is greater than $2$ (both are essential) then $w_i\neq -2$

Stick Condition
1. On any stick the weights of the vertices are non-zero except for end vertices that have a bond free in $\Gamma$ and for the case $\Gamma$ is:

$$\text{or}$$

2. The non-zero weights along any stick are of alternating sign. 3. No end vertex of a stick has weight $\pm 1$ unless it has a bond free in $\Gamma$.

II. If the valence of $v_i$ is $2$ then all of:

(Stick Condition)
1. $w_i\neq 0$
2. If valence of the parent or valence of the child is greater than $2$ (is essential) then $w_i\neq \pm1$
3. $\ $
  1. If valence of the parent is $2$ then $\text{sign}\LP w_i\RP\neq\text{sign}\LP w_{i-1}\RP$ (the parent)
  2. If valence of the child is $2$ then $\text{sign}\LP w_i\RP\neq\text{sign}\LP w_{i+1}\RP$ (the child)

Positivity Condition: Except for those with a free bond, there are no sticks in $\Gamma$ of the forms

$$\text{or}$$

II. If the valence of $v_i$ is $2$ then all of:

(Positivity Condition)
1. If valence of the parent and valence of the child is greater than $2$ (both are essential) then $w_i\neq -2$

Theorem [S]

$\Gamma$ is a $\LP+\RP$-CWPTT if and only if all the vertices of $\Gamma$ are $\LP+\RP$-canonical.

Tabulation of Arborescent Tangles

What do we have?

A combinatorial structure

A combinatorial structure
A way to distinguish two arborescent tangles

What we don’t have?

A unique representative

A unique representative
An efficient storage strategy

A unique representative
An efficient storage strategy
An efficient generation strategy

A unique representative

Definition [S]

A CWPTT is called right leaning if all weights are in the highest indexed region of each vertex. Additionally, any ring subtrees that are children of a vertex are the highest indexed children of that vertex.

Note

Not using “abbreviated trees” makes generation a bit less complex.

Theorem [S]

Every arborescent tangle has a right leaning CWPTT representative.

Definition [S]

A CWPTT is called an identity tree if its free bond is marked by $\iota\in V_4$.

Theorem [S]

Every arborescent tangle has an identity CWPTT representative.

Definition [S]

A CWPTT is called a right leaning identity tangle tree (RLITT) if it’s a right leaning and identity tree.

Theorem [S]

Every $\LP+\RP$-CWPTT has a unique right leaning identity representative.

✓ A unique representative
An efficient storage strategy
An efficient generation strategy

An efficient storage strategy

Cayley, A. (1857). ON THE THEORY OF THE ANALYTICAL FORMS CALLED TREES (The collected mathematical papers Vol. 3). Cambridge University Press. https://doi.org/10.1017/CBO9780511703690

✓ A unique representative
✓ An efficient storage strategy
An efficient generation strategy

An efficient generation strategy

Definition [S]

For WPTT $\Gamma_r$ and $\Gamma_s$, define the grafting operation $$ \begin{aligned} \Gamma_r\times\Gamma_s&\mapsto\Gamma_r\star_i\Gamma_s \end{aligned} $$ as follows. At the vertex $v_i$ of a $\Gamma_r$, introduce a bond connecting to the free bond at the root of $\Gamma_s$, reindexing as needed. We also require that $\Gamma_s$ be grafted so that the rightmost weight of $v_i$ and any ring subtrees of $v_i$ remain to the right of the scion after grafting.

When grafting at the root $v_0$ we omit the index label in the grafting operation, that is, $\star_0$ is written simply as $\star$. We call $\Gamma_r$ the rootstock and $\Gamma_s$ the scion.

$$\Gamma_r\text{ and }\color{#ffb86c}\Gamma_s$$

$$\Gamma_r\star_2\color{#ffb86c}\Gamma_s$$

Theorem [S]

Every $\Gamma$ $\LP+\RP$-RLITT of TCN $n$ is one of two forms:

$\Gamma$ is a single vertex with weight $\pm n$.
$\Gamma$ is the result of grafting at the root of some rootstock $\Gamma_r$ and $\LP+\RP$-RLITT scion $\Gamma_s$ where:
1. In $\Gamma_r$, $v_0$ is valence two, and $v_1$ is canonical except for violating the stick condition by $\text{Sign}\LP v_0\RP=\text{Sign}\LP v_1\RP$. Each vertex in $\LS v_i\RS_{i=2}^n$ of $\Gamma_r$ is $\LP +\RP$-canonical.
2. $\Gamma_r$ is $\LP+\RP$-RLITT

Algorithm [S]

Input

A collection of of RLITT scions $T_s$
A collection of RLITT rootstocks $T_r$

Output

A collection of weighted planar trees

Routine

for each combination of $\Gamma_r\in T_r$ and $\Gamma_s \in T_s$
1. Compute $\Gamma = \Gamma_r\star \Gamma_s$
2. Report $\Gamma$
3. Continue to the iteration of the loop if $v_0$ in $\Gamma_r$ is valence other than two
  1. Set $v_0$ to $-v_0$ for $\Gamma$
  2. Report $\Gamma$

Guarantee RLITT

Weight Condition
Identity Condition
Right Leaning Condition
Stick Condition
Positivity/Negativity Condition

Guarantee RLITT

✓ Weight Condition
✓ Identity Condition
✓ Right Leaning Condition
Stick Condition
Positivity/Negativity Condition

Stick Condition

Theorem [S]

For $\Gamma_r$ a $\LP+\RP-$RLITT and $\Gamma_s$ a $\LP+\RP-$RLITT scion, the result of $\Gamma=\Gamma_r\star\Gamma_s$ is canonical, if all $v_i$ at distance $1$ or less from the root are canonical.

Definition [S]

A $\LP+\RP-$RLITT (respectively $\LP-\RP-$RLITT) $\Gamma$ with root weight $w_0$ is called a good scion when either:

$w_0\neq0$
$w_0=0$ and the valence of $v_0$ is greater than $2$ (essential)

Algorithm [S]

Input

A collection of RLITT good scions $T_s$
A collection of RLITT rootstocks $T_r$

Output

A collection of weighted planar trees (still not guaranteed to be RLITT)

Routine

for each combination of $\Gamma_s\in T_s$ and $\Gamma_r \in T_r$
1. Compute $\Gamma = \Gamma_r\star \Gamma_s$
2. for each vertex $v_i$ at distance 1 from the root of $\Gamma$
  1. Continue to the next iteration of the outer loop if $v_i$ fails to satisfy the stick condition
3. Report $\Gamma$
4. Continue to the iteration of the loop if $v_0$ in $\Gamma_r$ is valence other than two
  1. Set $v_0$ to $-v_0$ for $\Gamma$
  2. Report $\Gamma$

Guarantee RLITT

✓ Weight Condition
✓ Identity Condition
✓ Right Leaning Condition
✓ Stick Condition
Positivity/Negativity Condition

Positivity/Negativity Condition

Theorem [S]

For $\Gamma_r$ a non-negative $\LP+\RP-$RLITT, and $\Gamma_s$ a good non-positive $\LP-\RP-$RLITT scion, the result of $\Gamma=\Gamma_r\star\Gamma_s$ is non-canonical.

Algorithm: Generate $\LP+\RP$-RLITT [S]

Input

A collection of $\LP+\RP$-RLITT good scions $T_s$
A collection of $\LP+\RP$-RLITT rootstocks $T_r$

Output

A collection of weighted planar trees (still not guaranteed to be RLITT)

Routine

for each combination of $\Gamma_s\in T_s$ and $\Gamma_r \in T_r$
1. Compute $\Gamma = \Gamma_r\star \Gamma_s$
2. for each vertex $v_i$ at distance 1 from the root of $\Gamma$
  1. Continue to the next iteration of the outer loop if $v_i$ fails to satisfy the stick condition
  2. Continue to the next iteration of the outer loop if $v_i$ fails to satisfy the positivity condition
3. Report $\Gamma$
4. Continue to the iteration of the loop if $v_0$ in $\Gamma_r$ is valence other than two
  1. Set $v_0$ to $-v_0$ for $\Gamma$
  2. Report $\Gamma$

Algorithm: Generate $\LP-\RP$-RLITT [S]

Input

A collection of $\LP-\RP$-RLITT good scions $T_s$
A collection of $\LP-\RP$-RLITT rootstocks $T_r$

Output

A collection of weighted planar trees (still not guaranteed to be RLITT)

Routine

for each combination of $\Gamma_s\in T_s$ and $\Gamma_r \in T_r$
1. Compute $\Gamma = \Gamma_r\star \Gamma_s$
2. for each vertex $v_i$ at distance 1 from the root of $\Gamma$
  1. Continue to the next iteration of the outer loop if $v_i$ fails to satisfy the stick condition
  2. Continue to the next iteration of the outer loop if $v_i$ fails to satisfy the positivity condition
3. Report $\Gamma$
4. Continue to the iteration of the loop if $v_0$ in $\Gamma_r$ is valence other than two
  1. Set $v_0$ to $-v_0$ for $\Gamma$
  2. Report $\Gamma$

Guarantee RLITT

✓ Weight Condition
✓ Identity Condition
✓ Right Leaning Condition
✓ Stick Condition
✓ Positivity/Negativity Condition

Partitioning jobs

$$\begin{aligned} (0&,\text{TCN})\\ ( 1&,\text{TCN}-1)\\ &\vdots\\ (\text{TCN}-1&,1)\\ (\text{TCN}&,0) \end{aligned} $$

Algorithm [S]

Input

A target TCN $n$

Output

A set $T$ of all RLITT up to TCN

Routine

Set $T$ to be the set $\LS \iota[0],\ \iota[0\ 0],\ \iota[1],\ \iota[-1],\ \iota[2],\ \iota[-2],\ \cdots,\ \iota[n],\ \iota[-n],\ \RS$
for i from 2 to TCN
1. for j from i-2 to 1
  1. Set $T_{r+}$ to be the set of $(+)$-RLITT with TCN equal to $j$
  2. Set $T_{r-}$ to be the set of $(-)$-RLITT with TCN equal to $j$
  3. Set $T_{s+}$ to be the set of $(+)$-RLITT good scions with TCN equal to $n-j$
  4. Set $T_{s-}$ to be the set of $(-)$-RLITT good scions with TCN equal to $n-j$
  5. Execute “Generate $\LP+\RP$-RLITT” input $T_{r+}$ and $T_{s+}$
  6. Execute “Generate $\LP-\RP$-RLITT” input $T_{r-}$ and $T_{s-}$
2. Set $T_{r+}$ to be the set of $(+)$-RLITT with TCN equal to $0$
3. Set $T_{r-}$ to be the set of $(-)$-RLITT with TCN equal to $0$
4. Set $T_{s+}$ to be the set of $(+)$-RLITT good scions with TCN equal to $i$
5. Set $T_{s-}$ to be the set of $(-)$-RLITT good scions with TCN equal to $i$
6. Execute “Generate $\LP+\RP$-RLITT” input $T_{r+}$ and $T_{s+}$
7. Execute “Generate $\LP-\RP$-RLITT” input $T_{r-}$ and $T_{s-}$
8. Add the results to $T$

Technologies

Future work

Minimalization

of Arborescent representatives

$$\leftarrow$$

Minimalization

general representatives

Tabulation of Polygonal tangles

Algebraic

$+$, $\vee$, and plumbing only form bigons between basic tangles in the “knot shadow”.

Polygonal tangles

How to form something other than bigons?

$6^{\ast\ast}$

Minimally polygonal arborescent tangles

Solution?

Tabulate the polygonal tangles.

Thank you!

Special thanks to the committee members:

Dr. Isabel Darcy, Supervisor
Dr. Francis Bonahon
Dr. Keiko Kawamuro
Dr. Colleen Mitchell
Dr. Radmila Sazdanović

Refernces

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
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.
Robert Glenn Scharein. Interactive topological drawing. ProQuest LLC, Ann Arbor, MI, 1998. Thesis (Ph.D. The University of British Columbia (Canada). URL: https://www.knotplot.com/.
Jablan, S., & Sazdanović, R. (2007). Linknot. In Series on Knots and Everything. WORLD SCIENTIFIC. https://doi.org/10.1142/6623
Dowker, C. H., & Thistlethwaite, M. B. (1983). Classification of knot projections. In Topology and its Applications (Vol. 16, Issue 1, pp. 19-31). Elsevier BV. https://doi.org/10.1016/0166-8641(83)90004-4
Hoste, J., Thistlethwaite, M., & Weeks, J. (1998). The first 1,701,936 knots. In The Mathematical Intelligencer (Vol. 20, Issue 4, pp. 33-48). Springer Science and Business Media LLC. https://doi.org/10.1007/bf03025227
Burton, B. A. (2020). The Next 350 Million Knots. Schloss Dagstuhl - Leibniz-Zentrum Für Informatik. https://doi.org/10.4230/LIPICS.SOCG.2020.25
Connolly, Nicholas. Classification and Tabulation of 2-String Tangles: The Astronomy of Subtangle Decompositions. University of Iowa, 2021, https://doi.org/10.17077/etd.005978.
Cayley, A. (2009). The collected mathematical papers (Vol. 3). Cambridge University Press. https://doi.org/10.1017/CBO9780511703690
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

Refernces

Facebook, Public domain, via Wikimedia Commons
FastAPI The MIT License (MIT)
Carlos Baraza, CC0, via Wikimedia Commons
Qq1040058283, Public domain, via Wikimedia Commons
Jeremy Kratz, Public domain, via Wikimedia Commons
Cython and Python, Apache License 2.0, via Wikimedia Commons
mermaidjs
www.python.org, GPL, via Wikimedia Commons
Mongodb
Ryan Dahl, MIT, via Wikimedia Commons
Holger Krekel, CC BY 2.5, via Wikimedia Commons
Alon Zakai, MIT, via Wikimedia Commons
Cmake team. The original uploader was Francesco Betti Sorbelli at Italian Wikipedia.. Vectorized by Magasjukur2, CC BY 2.0, via Wikimedia Commons
Emoji One, CC BY-SA 4.0, via Wikimedia Commons