Use Case

Functionality

An algebraic tangle is any tangle generated from an "algebraic" statement built from the two binary operations \(+\) and \(\vee\) on the four basic tangles (\(0,\ \pm1,\ \infty\)).

\(\LP\LP\infty+\infty\RP\vee\LP 0\vee0\RP\RP+\LP1+0\RP+\infty\)

Since each of \(+\) and \(\vee\) are binary operations we can interpret these algebraic statements as binary trees where each non-leaf vertex corresponds to an operation and each leaf a basic tangle, we call these algebraic tangle trees. The concept of a algebraic tree decomposition was given first by Caudron ¹ and then further refined by Connolly ².

Example

flowchart TD

  id0("+")-->id1("∞")
  id0("+")-->id2("∞")

  id6("+")-->id7("1")
  id6("+")-->id8("0")

  id3("v")-->id4("0")
  id3("v")-->id5("0")

  id9("v")-->id0
  id9("v")-->id3

  id10("+")-->id9
  id10("+")-->id6
  id11("+")-->id10
  id11("+")-->id12("∞")

To simplify the combinatorics we can substitute rational tangles (as) twist vectors for the basic tangles and record the tree as a string in polish notation

Example

Algebraically:

\([1 2 0]+\LP[2 1 0]+[2 2 0]\RP\)

In polish notation:

\(+[1 2 0]+[2 1 0][2 2 0]\)

As a algebraic tangle tree att "decoded":

flowchart TD
id0("+")-->id1("[1 2 0]")
id0("+")-->id2("+")
id2("+")-->id3("[2 1 0]")
id2("+")-->id4("[2 2 0]")

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1. A. Caudron. Classification Des Noeuds et Des Enlacements. Prépublications / Université de Paris-Sud, Département de Mathématiques. Université de Paris-Sud, Dép. de mathématique, 1982. URL: https://books.google.com/books?id=W1nvAAAAMAAJ. ↩

2. Nicholas Connolly. Classification and Tabulation of 2-String Tangles: The Astronomy of Subtangle Decompositions. University of Iowa, 2025. URL: https://iro.uiowa.edu/esploro/outputs/doctoral/9984124571002771 (visited on 2023-05-04), doi:10.17077/etd.005978. ↩