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Classification of algebraic tangles

Bartosz Ambrozy Gren, Joanna Ida Sulkowska, Boštjan Gabrovšek

TL;DR

The paper tackles the classification of algebraic tangles by introducing a binary-tree, canonical representation that fixes twist, flype, ring, and elementary moves. It extends the catalog of prime tangles up to $14$ crossings, distinguishes mutants, and analyzes tangle symmetry groups, supported by a public database and open-source code. The approach combines a rigorous tree-based decomposition with a generalized fraction invariant and a multi-stage canonicalization algorithm (treefix) to produce unique representations. This framework advances knot theory tooling for tabulating algebraic tangles and enables applications in biology and computational topology through scalable enumeration and symmetry analysis.

Abstract

We study algebraic tangles as fundamental components in knot theory, developing a systematic approach to classify and tabulate prime tangles using a novel canonical representation. The canonical representation enables us to distinguish mutant tangles, which fills the gaps in previous classifications. Moreover, we increase the classification of prime tangles up to 14 crossings and analyze tangle symmetry groups. We provide a database of our results: https://tangleinfo.cent.uw.edu.pl.

Classification of algebraic tangles

TL;DR

The paper tackles the classification of algebraic tangles by introducing a binary-tree, canonical representation that fixes twist, flype, ring, and elementary moves. It extends the catalog of prime tangles up to crossings, distinguishes mutants, and analyzes tangle symmetry groups, supported by a public database and open-source code. The approach combines a rigorous tree-based decomposition with a generalized fraction invariant and a multi-stage canonicalization algorithm (treefix) to produce unique representations. This framework advances knot theory tooling for tabulating algebraic tangles and enables applications in biology and computational topology through scalable enumeration and symmetry analysis.

Abstract

We study algebraic tangles as fundamental components in knot theory, developing a systematic approach to classify and tabulate prime tangles using a novel canonical representation. The canonical representation enables us to distinguish mutant tangles, which fills the gaps in previous classifications. Moreover, we increase the classification of prime tangles up to 14 crossings and analyze tangle symmetry groups. We provide a database of our results: https://tangleinfo.cent.uw.edu.pl.

Paper Structure

This paper contains 21 sections, 15 theorems, 35 equations, 17 figures, 8 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation and isotopy preserving moves
  4. An algebraic tangle as a tree
  5. Tree notation
  6. Right leaves -- integral tangles
  7. Left leaves -- rational tangles
  8. Fraction of an algebraic tangle
  9. Canonical representation
  10. Classification of prime tangles
  11. Generating tangles and the classification methods
  12. Generation of alternating tangles
  13. Generation of non-alternating tangles
  14. Determination of tangles' minimal set and symmetries
  15. Minimization algorithm
  16. ...and 6 more sections

Key Result

Theorem 1

Two tangles are isotopic if and only if their diagrams differ by planar isotopy and a finite sequence of Reidemeister moves.

Figures (17)

  • Figure 1: a) A Conway's sphere, in which the tangle is embedded. b) Basic tangles which satisfy the boundary. c) Sum and product of tangles R and F.
  • Figure 2: Reidemeister moves of Type I, II, and III.
  • Figure 3: Tangle $R$ transformed by rotations and reflections represented by a two-faced square with letter "R". a) Set of all tangles equivalent to tangle $R$ ordered by rotations: $\nu$ (horizontally), $\rho_y$ (vertically). b) Reflections of tangle $R$: $\eta R \equiv R0$, and $\mu R \equiv \overline{R}$.
  • Figure 4: Tangle moves: elementary moves I and II (special cases of Reidemeister I and II moves), twist, ring and flype moves. Two presented flypes are related by Reidemeister III move.
  • Figure 5: Examples of non-prime tangles $0(20)$ and $0(30)$ (respectively).
  • ...and 12 more figures

Theorems & Definitions (49)

  • Definition 1: Tangle
  • Definition 2: VHX
  • Definition 3: Basic tangle
  • Definition 4: Sum and product of tangles
  • Definition 5: Algebraic tangle
  • Definition 6: Isotopy
  • Definition 7: Equivalence
  • Definition 8: Tangle diagram
  • Theorem 1: conway1970enumeration
  • Definition 9: Rotations and reflections
  • ...and 39 more