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Yarn Ball Knots and Faster Computations

Dror Bar-Natan, Itai Bar-Natan, Iva Halacheva, Nancy Scherich

TL;DR

The paper investigates whether knot invariants can be computed more efficiently in 3D than in the traditional 2D diagrammatic setting. By modeling knots as yarn balls and grid knots, it establishes a 3D input size $V$ and compares with the 2D projected size $n \sim V^{4/3}$, proving the linking number is computable in time $C_{lk}(3D,V) \sim V$ whereas 2D methods require $C_{lk}(2D,n) \sim n$. It generalizes to finite type invariants of type $d$, showing $C_{\zeta}(3D,V) \lesssim V^d$ and $C_{\zeta}(2D,n) \lesssim n^d$, using Gauss-diagram techniques and a detailed combinatorial counting framework. The results provide a foundational 3D computational advantage and outline a program to extend 3D methods to broader knot invariants and quantities, with concrete conjectures and counting lemmas to guide future improvements.

Abstract

We make use of the 3D nature of knots and links to find savings in computational complexity when computing knot invariants such as the linking number and, in general, most finite type invariants. These savings are achieved in comparison with the 2D approach to knots using knot diagrams.

Yarn Ball Knots and Faster Computations

TL;DR

The paper investigates whether knot invariants can be computed more efficiently in 3D than in the traditional 2D diagrammatic setting. By modeling knots as yarn balls and grid knots, it establishes a 3D input size and compares with the 2D projected size , proving the linking number is computable in time whereas 2D methods require . It generalizes to finite type invariants of type , showing and , using Gauss-diagram techniques and a detailed combinatorial counting framework. The results provide a foundational 3D computational advantage and outline a program to extend 3D methods to broader knot invariants and quantities, with concrete conjectures and counting lemmas to guide future improvements.

Abstract

We make use of the 3D nature of knots and links to find savings in computational complexity when computing knot invariants such as the linking number and, in general, most finite type invariants. These savings are achieved in comparison with the 2D approach to knots using knot diagrams.

Paper Structure

This paper contains 10 sections, 6 theorems, 19 equations, 12 figures.

Table of Contents

  1. Introduction
  2. Motivation
  3. Yarn balls and pancakes
  4. Computational complexities of knot invariants
  5. 2D versus 3D algorithms
  6. Further directions
  7. Grid Knots and Linking Number
  8. Linking Number
  9. Finite Type Invariants
  10. Combinatorial Results

Key Result

Theorem 2.1

$C_{lk}(3D,V)\sim V$, while $C_{lk}(2D,n)\sim n$, so $lk$ is C3D.

Figures (12)

  • Figure 1: (A) A planar diagram of a knot in a pancake (knot diagram by Piccirillo Piccirillo). (B) A yarn ball.
  • Figure 2: Projection of a yarn ball knot to a disk of diameter $L$. To count the crossings in the projection, subdivide the disk into $1\times 1$ squares. Each square will have $\sim \frac{L}{2}$ strands crossing each other, for a total of $\sim {L \choose 2} \sim L^2$ crossings in that square.
  • Figure 3: Two examples of grid knots. The left grid knot has $L=3$ and 64 labeled arcs, and the right has $L=5$ and 216 labeled arcs.
  • Figure 4: The grid in (B) shows a slightly askew top down view of the grid from (A). (C) highlights two crossing fields, $F_1$ and $F_2$, of a grid.
  • Figure 5: The 8 crossing types, where "/" strands are green and "\\" are red. For later use we label these crossings $x_1$ through $x_8$.
  • ...and 7 more figures

Theorems & Definitions (15)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Definition 1.4
  • Theorem 2.1
  • proof
  • Theorem 3.1: Goussarov-Polyak-Viro GPV, see also Roukema
  • Theorem 3.2
  • Theorem 3.3
  • proof
  • ...and 5 more