Alternating knots with large boundary slope diameter

TL;DR

This paper demonstrates that for alternating knots, the ratio of the boundary slope diameter to the crossing number can be made arbitrarily large, revealing a significant variability in boundary slope behavior.

Contribution

It establishes that the boundary slope diameter to crossing number ratio in alternating knots is unbounded, providing new insights into their geometric properties.

Findings

Ratio of boundary slope diameter to crossing number can be arbitrarily large for alternating knots.

Boundary slope diameter exhibits significant variability relative to crossing number.

Results challenge previous assumptions about boundary slope bounds in knot theory.

Abstract

We show that, for an alternating knot, the ratio of the diameter of the set of boundary slopes to the crossing number can be arbitrarily large.