Geometric learning of knot topology

Area of Science:

• Knot theory
• Computational topology
• Machine learning

Background:

• Distinguishing complex knots is a major challenge in knot theory.
• The relationship between a curve's geometry and its topology is not fully understood.
• Existing knot invariants can be insufficient for precise classification.

Purpose of the Study:

• To develop a neural network (NN) approach for predicting knot topology from geometric data.
• To investigate if geometric features can serve as robust knot invariants.
• To explore the potential of NNs in classifying and localizing knots.

Main Methods:

• Utilizing a neural network (NN) trained on geometric representations of knotted curves.
• Employing a "local writhe" feature for knot characterization.
• Scaling the NN approach to classify prime knots up to 10-crossings.
• Training NNs for knot localization tasks.

Main Results:

• NNs trained with "local writhe" accurately distinguish knots with shared topological invariants.
• The NN approach outperforms some traditional knot polynomials in classifying complex knots like mutants and composites.
• High accuracy (over 95%) achieved in classifying prime knots up to 10-crossings.
• NNs successfully applied to knot localization problems.

Conclusions:

• The "local writhe" pattern serves as a potent geometric signature for knot topology.
• This NN-based method offers a powerful new tool for knot classification and analysis.
• The findings suggest potential applications in soft matter physics and the development of new topological invariants.