Knots with equal bridge index and braid index

TL;DR

This paper investigates BB knots, where the bridge index equals the braid index, exploring their geometric properties, identifying examples within specific knot families, and analyzing their growth with crossing number.

Contribution

It provides a heuristic explanation and numerical exploration of the conjecture that BB knots have a near-planar equilibrium configuration when realized as springy wires.

Findings

Identified all BB knots with up to 12 crossings in certain families.

Found exponential growth in the number of BB knots with crossing number.

Supported the conjecture with numerical evidence and family classifications.

Abstract

In this paper we are interested in BB knots, namely knots and links where the bridge index equals the braid index. Supported by observations from experiments, it is conjectured that BB knots possess a special geometric/physical property (and might even be characterized by it): if the knot is realized by a (closed) springy metal wire, then the equilibrium state of the wire is in an almost planar configuration of multiple (overlapping) circles. In this paper we provide a heuristic explanation to the conjecture and explore the plausibility of the conjecture numerically. We also identify BB knots among various knot families. For example, we are able to identify all BB knots in the family of alternating Montesinos knots, as well as some BB knots in the family of the non-alternating Montesinos knots, and more generally in the family of the Conway algebraic knots. The BB knots we identified in the knot families we considered include all of the 182 one component BB knots with crossing number up to 12. Furthermore, we show that the number of BB knots with a given crossing number $n$ grows exponentially with $n$.