U of M - Dearborn Colloquium (11/15/23)

The Tanglenomicon

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

Mathematics Department at The University of Iowa

Knots

“A knot is a smooth embedding of a circle $S^1$ into Euclidean 3-dimensional space $\R^3$ (or the 3-dimensional sphere $S^3$ ).”

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Jablan, S., & Sazdanović, R. (2007). Linknot. In Series on Knots and Everything. WORLD SCIENTIFIC. https://doi.org/10.1142/6623

https://www.knotplot.com/

Knot Equivalence

reidemeister moves

Type I

$\leftrightarrow$

Type II

$\leftrightarrow$

Type III

$\leftrightarrow$

The natural question

Knot Tables

Lord Kelvin’s vortex theory of the atom. Atoms are knotted vortices in the æther.
  • 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

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. “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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NWNESWSE
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Basic Operations

Operation $+$

$+$
$=$
$=$
$=$
$2$

Operation $\vee$

$\vee$
$=$
$=$
$=$
$\frac{1}{2}$

The Tanglenomicon

Building up

$\ $
$\ $
$\ $
$\ $

Where we are

Rational Tangles

$\ $
$\begin{aligned}\to&\ \LP 3 \vee \frac{1}{2}\RP + 2\\&\\ \to&\ [3\ 2\ 2]\end{aligned}$

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 $0\to 2^{N-1}$.

Twist Vectors for $N=5$
$$\begin{array}{|l|l|l|l|} \hline [1\ 1\ 1\ 1\ 1]\ &\ [2\ 1\ 1\ 1]\ &\ [1\ 2\ 1\ 1]\ &\ [1\ 1\ 2\ 1]\\\hline [1\ 1\ 1\ 2]\ &\ [3\ 1\ 1]\ &\ [1\ 3\ 1]\ &\ [1\ 1\ 3]\\\hline [2\ 2\ 1]\ &\ [2\ 1\ 2]\ &\ [1\ 2\ 2]\ &\ [3\ 2]\\\hline [2\ 3]\ &\ [4\ 1]\ &\ [1\ 4]\ &\ [5]\\\hline \end{array}$$

Canonical Twist Vectors

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

Canonical Twist Vectors for $N=5$
$$\begin{array}{|l|l|l|l|} \hline [1\ 1\ 1\ 1\ 1]\ &\ [2\ 1\ 1\ 1\ 0]\ &\ [1\ 2\ 1\ 1\ 0]\ &\ [1\ 1\ 2\ 1\ 0]\\\hline [1\ 1\ 1\ 2\ 0]\ &\ [3\ 1\ 1]\ &\ [1\ 3\ 1]\ &\ [1\ 1\ 3]\\\hline [2\ 2\ 1]\ &\ [2\ 1\ 2]\ &\ [1\ 2\ 2]\ &\ [3\ 2\ 0]\\\hline [2\ 3\ 0]\ &\ [4\ 1\ 0]\ &\ [1\ 4\ 0]\ &\ [5]\\\hline \end{array}$$
Programmatic Description
stateDiagram-v2 direction LR state if_done <<choice>> State_i: i=0 State_ipp: i++ state "Construct TV from i as a bitfield" as tv_calc{ state "tmp=i;j=0;cnt=N" as State_temp State_jpp: j++ State_cntmm: cnt-- State_sum_tv: TV[j]++ State_rsh: tmp=tmp>>1 state if_lsb <<choice>> state if_cnteo <<choice>> State_store_tv: Store TV [*] --> State_temp State_temp --> if_cnteo if_cnteo--> State_cntmm: if cnt>0 if_cnteo--> State_store_tv: if cnt==0 State_store_tv --> [*] State_cntmm -->if_lsb if_lsb -->State_sum_tv: if (tmp & 0x01u)==1u State_sum_tv --> State_rsh if_lsb -->State_jpp: if (tmp & 0x01u)==0u State_jpp --> State_rsh State_rsh --> if_cnteo } [*] --> State_i State_i --> if_done if_done --> tv_calc: if i < 2**(N-1) tv_calc --> State_ipp State_ipp --> if_done if_done --> [*]: if i == 2**(N-1)

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: $$\LB a\ b\ c\RB=c+\frac{1}{b+\frac{1}{a}}$$

Twist Vector to rational number
$$\ =\LB 3\ 2\ 2\RB=2+\frac{1}{2+\frac{1}{3}}=\frac{17}{7}$$

Louis H. Kauffman and Sofia Lambropoulou. Classifying and applying rational knots and rational tangles. In DeTurck, editor, Contemporary Mathematics, volume 304, pages 223-259, 2001

To play with twist vectors and continued fractions visit