Boundary slopes of 2-bridge links determine the crossing number

TL;DR

This paper establishes a relationship between the boundary slopes of diagonal surfaces in 2-bridge link exteriors and the crossing number, providing a new way to determine the crossing number from boundary slope data.

Contribution

It derives a formula for boundary slopes of diagonal surfaces in 2-bridge link exteriors and proves that the link's crossing number equals the diameter of these slopes.

Findings

Boundary slope formula for 2-bridge links

Crossing number equals the diameter of boundary slopes

Extension of Hatcher and Thurston's knot results

Abstract

A diagonal surface in a link exterior M is a properly embedded, incompressible, boundary incompressible surface which furthermore has the same number of boundary components and same slope on each component of the boundary of M. We derive a formula for the boundary slope of a diagonal surface in the exterior of a 2-bridge link which is analogous to the formula for the boundary slope of a 2-bridge knot found by Hatcher and Thurston. Using this formula we show that the diameter of a 2-bridge link, that is, the difference between the smallest and largest finite slopes of diagonal surfaces, is equal to the crossing number.