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 the 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.
Permission has been granted by Kobe Journal of Mathematics to supply this article for educational and research purposes. More info can be found about the Kobe Journal of Mathematics at http://www.math.kobe-u.ac.jp/jmsj/kjm/. © Kobe Journal of Mathematic.
Hoste, Jim and Patrick Shanahan. Boundary slopes of 2-bridge links determine the crossing number. Kobe Journal of Mathematics, Vol. 24 (2007) 21-39.