A table of knotoids in $S^3$ up to seven crossings

This paper presents a complete classification of spherical knotoids with up to six crossings and a conjectured complete classification up to seven crossings, utilizing a suite of invariants to distinguish equivalence classes while exploring their symmetries and applications to protein entanglement.

The Gist

Imagine you have a piece of string with two loose ends. In the world of classical knot theory, mathematicians usually tie the two ends together to make a loop, like a rubber band, so they can study the "knot" inside. But in the real world, many things aren't loops. Think of a protein in your body or a strand of DNA; they are long chains with distinct start and end points.

This paper is about a new way of studying those open-ended tangles, called knotoids. The authors, Boštjan Gabrovšek and Paolo Cavicchioli, have created a massive "phone book" or catalog of these tangles, sorting them by how complex they are (measured by the number of times the string crosses over itself). They have successfully cataloged every unique tangle up to six crossings and made a very strong guess about the ones with seven crossings.

Here is a simple breakdown of what they did and why it matters:

1. The Problem: The "Forbidden Move"

In normal knot theory, if you have a loop, you can slide a piece of the rope over the top of the loop to untangle it. But with a knotoid (an open string), you cannot slide the ends over the rest of the string. The ends are "stuck" in place. This restriction means that many tangles that would look like simple loops to a classical knot theorist are actually complex, knotted structures when the ends are fixed.

Analogy: Imagine a snake sliding through a garden hose. If the hose is a loop, the snake can wiggle out easily. But if the hose is open at both ends and pinned to the ground, the snake might get hopelessly tangled even if it looks simple from the outside.

2. The Mission: Building the Catalog

The authors wanted to create a complete list of all possible "prime" knotoids (the basic building blocks that can't be broken down into smaller knots) with up to seven crossings.

• Step 1: The Generator. They used a computer to generate millions of possible diagrams of these tangles.
• Step 2: The Simplifier. They used a set of rules (called Reidemeister moves) to try and untangle the diagrams. If a diagram could be simplified into a smaller one, they threw it out. They were looking for the "minimal" version of each knot.
• Step 3: The Fingerprint. This is the hardest part. Two tangles might look different but are actually the same. To tell them apart, the authors used a collection of mathematical "fingerprints" (polynomials).
  • The Kauffman Bracket & Arrow Polynomial: These are like checking the DNA of the knot.
  • The Yamada Polynomial: This is the "super-fingerprint." It's the most powerful tool they used, but it's also the most computationally expensive (like running a super-complex simulation).
  • The Mock Alexander & Affine Index: These are other specialized tests to catch subtle differences.

3. The Results: The Big Numbers

After running millions of calculations, they found:

• 427 distinct types of prime spherical knotoids up to 7 crossings.
• They identified which ones are Chiral (they have a "left-handed" and "right-handed" version, like your hands) and which are Rotatable (they look the same if you spin them 180 degrees).
• They found 14 pairs that were so similar that their current mathematical fingerprints couldn't tell them apart. They suspect these are "mutant" pairs—twins that are mathematically identical in every test they ran, but might be slightly different in a way we haven't discovered yet.

4. Why Should You Care? (The Protein Connection)

You might wonder, "Who cares about open string tangles?" The answer is biology.

Proteins are the workhorses of your body. They are long chains of amino acids that fold up into complex 3D shapes. Sometimes, these chains get knotted.

• The Old Way: To study a knotted protein, scientists used to pretend the ends were tied together. But this is artificial; it changes the shape and can hide the true nature of the knot.
• The New Way: By using knotoids, scientists can study the protein exactly as it is—an open chain. This helps them understand how proteins fold, how they get stuck, and how diseases related to protein misfolding (like Alzheimer's) might work.

Summary

Think of this paper as the first comprehensive dictionary of open-string knots. Just as a dictionary helps you understand the building blocks of language, this table helps biologists and mathematicians understand the building blocks of entanglement in nature. They have mapped out the landscape of these tangles up to a certain complexity, providing a foundation for future discoveries in both mathematics and molecular biology.

The authors admit that for the very last few entries (the 7-crossing ones), they hit a wall where their current tools couldn't distinguish between twins. But they have laid the groundwork, and their "phone book" is now available for anyone to use to decode the tangled secrets of the universe.

Technical Summary

Here is a detailed technical summary of the paper "A table of knotoids in $S^3$ up to seven crossings" by Boštjan Gabrovšek and Paolo Cavicchioli.

1. Problem Statement

The paper addresses the classification of spherical knotoids, a generalization of classical knots introduced by V. Turaev. Unlike classical knots, which are closed loops, knotoids are equivalence classes of generic immersions of an oriented unit interval $[0, 1]$ into a surface (specifically the 2-sphere $S^2$ in this study).

The core challenge in knotoid tabulation is that the complement of a knotoid is not a 3-manifold, meaning the powerful tools of hyperbolic geometry (used extensively in classical knot tabulation, such as computing canonical hyperbolic triangulations) are unavailable. Consequently, distinguishing between distinct knotoids requires relying solely on combinatorial simplifications (Reidemeister moves) and algebraic invariants. Previous work had classified knotoids up to six crossings, but a complete, independent classification up to seven crossings was lacking, particularly regarding the resolution of "mutant-type" pairs where standard invariants fail to distinguish chirality or rotational symmetry.

2. Methodology

The authors employed a multi-step computational pipeline to generate, simplify, and classify prime spherical knotoids.

• Diagram Generation:

  • Generated all non-isomorphic planar graphs with connectivity 1 and minimal degree 1 using plantri software.
  • Filtered for graphs with exactly two degree-1 vertices (endpoints) and $n-2$ degree-4 vertices (crossings).
  • Converted these graphs into knotoid diagrams by assigning positive or negative crossing types to every 4-degree vertex.
  • This initial step yielded 160,825 distinct canonical EM (Ewing-Millett) codes.

• Simplification and Filtering:

  • Applied simple crossing-reducing Reidemeister moves (I and II) to reduce the set to 28,447 diagrams.
  • Removed composite knotoids (connected sums) and concatenations, leaving 11,612 and 1,163 respectively.
  • Performed brute-force simplification to find minimal representatives. The process involved systematic application of Reidemeister moves, including:
    • Moves that do not increase crossings.
    • Moves involving one crossing-increasing Reidemeister II move.
    • Moves involving flypes (180° rotations of tangles).
  • This reduced the candidate set to 2,600 diagrams.

• Invariant Computation and Partitioning:

  • Computed a suite of polynomial invariants for the remaining candidates:
    • Kauffman Bracket Polynomial
    • Arrow Polynomial (extended from virtual knots)
    • Mock Alexander Polynomial (adapted for spherical knotoids by "starring" a region)
    • Affine Index Polynomial
    • Yamada Polynomial of the double-sided closure (viewing the knotoid closure as a $\Theta$-curve).
  • The Yamada polynomial was identified as the most computationally expensive but most powerful invariant.
  • This step partitioned the 2,600 diagrams into 49 unique groups and 772 groups of diagrams sharing identical invariant values.

• Exhaustive Search and Symmetry Analysis:

  • For groups with identical invariants, the authors performed simultaneous exhaustive searches of the Reidemeister move space. They checked if the search trees of two diagrams met at a common intermediate diagram (indicating equivalence).
  • To address rotational symmetry, they expanded the search space to include the rotation of the knotoid ($rot(K)$) within the equivalence check.
  • Chirality Analysis: Mirror images were computed. If a knotoid $K$ and its mirror $K^*$ shared invariants and could not be distinguished by Reidemeister moves, they were treated as a chiral pair.

3. Key Contributions

• Complete Classification up to 6 Crossings: The paper provides an independent and verified complete classification of prime spherical knotoids with up to six crossings.
• Conjectured Classification up to 7 Crossings: The authors present a classification up to seven crossings. While they successfully distinguished the vast majority of cases, 14 pairs of diagrams (forming 28 groups) remain unverified as distinct due to the limitations of current invariants.
  • The authors conjecture these 14 pairs are non-rotatable knotoids that are indistinguishable by the computed invariants, analogous to "mutant" knots in classical theory.
• Invariance Performance Analysis: The study quantitatively evaluates the distinguishing power of various invariants, showing that the Yamada polynomial of the closure is the most effective tool, followed closely by the Arrow and Kauffman bracket polynomials. The Affine index polynomial was found to have zero distinguishing power for the final set of candidates.
• Open-Source Tools: The authors utilized and contributed to the KnotPy Python library, making the source code, results, and full tabulation (including PD and EM notations) publicly available.

4. Results

The computational enumeration produced the following statistics (Table 1 in the paper):

• Total Prime Spherical Knotoids: 427 distinct types identified up to 7 crossings.
• Crossing Distribution:
  • 0 crossings: 1
  • 1 crossing: 0
  • 2 crossings: 1
  • 3 crossings: 2
  • 4 crossings: 8
  • 5 crossings: 25
  • 6 crossings: 82
  • 7 crossings: 308
• Symmetry Properties:
  • Chirality: 394 knotoids are chiral, while 33 are achiral.
  • Rotatability: 37 knotoids are rotatable (symmetric under 180° rotation), while 390 are non-rotatable.
  • Fully Achiral and Rotatable: Only three knotoids (01, 41, 63) possess both properties; these correspond to fully achiral classical knots.
• Ambiguities: There are 14 potential duplicate pairs (28 knotoids) where invariants fail to distinguish the knotoid from its mirror or rotation. These are flagged as "Possible duplicates" in the appendix (e.g., $K7_{26}$ and $K7_{28}$).

5. Significance

• Foundational Resource: This table serves as a foundational reference for future research in knotoid theory, extending the tradition of knot tabulation (initiated by Tait) to open-ended entanglements.
• Protein Entanglement Applications: The classification is directly applicable to molecular biology. Proteins are open chains, and their topological entanglements can be modeled as knotoids. This table provides "fingerprints" (via invariants like the Yamada polynomial) to analyze protein backbones without the need for artificial closure, which can obscure topological features. Tools like Knoto-ID rely on such classifications.
• Computational Topology: The paper highlights the computational challenges of classifying open structures where hyperbolic geometry is inapplicable, demonstrating the necessity of combining brute-force simplification with a diverse suite of polynomial invariants.
• Theoretical Insight: The identification of potential "mutant-type" knotoid pairs suggests that the classification of open entanglements may face similar limitations to classical knot theory, where standard invariants cannot always distinguish between distinct topological structures.