Stair-step bridge spectra does not imply high distance

TL;DR

This paper demonstrates that having a stair-step bridge spectrum does not necessarily imply high distance in knots, by analyzing specific classes of Montesinos and pretzel knots.

Contribution

It provides the first examples of knots with stair-step bridge spectra that are not high distance, expanding understanding of the relationship between these properties.

Findings

Identified classes of knots with stair-step bridge spectra that are not high distance

Computed bridge spectra and distance for generalized Montesinos knots

Provided the first examples of such knots in the literature

Abstract

Tomova, along with results of Bachman and Schleimer, showed that any high distance knot has a stair-step bridge spectrum. In this paper, we compute the bridge spectra and distance of generalized Montesinos knots. In particular, we produce the first example of a class of knots which attain the stair-step bridge spectra but are not high distance, by computing the bridge spectra of certain pretzel knots and Montesinos knots.