Symmetric Criticality for Tight Knots

TL;DR

This paper establishes a symmetric criticality principle for ropelength-critical knots, enabling the construction of new critical configurations with symmetry that differ from minimal ropelength solutions.

Contribution

It proves a symmetric criticality theorem for ropelength-critical knots, allowing the generation of novel symmetric critical configurations beyond minimal ropelength solutions.

Findings

Symmetric criticality theorem for ropelength-critical knots.

Construction of new symmetric critical configurations.

Existence of critical configurations different from ropelength minima.

Abstract

We prove a version of symmetric criticality for ropelength-critical knots. Our theorem implies that a knot or link with a symmetric representative has a ropelength-critical configuration with the same symmetry. We use this to construct new examples of ropelength critical configurations for knots and links which are different from the ropelength minima for these knot and link types.