Boundary slopes of 2-bridge links determine the crossing number

Jim E. Hoste; Patrick D. Shanahan

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Boundary slopes of 2-bridge links determine the crossing number

Jim E. Hoste·Patrick D. Shanahan

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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.

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Jim E. Hoste, Patrick D. Shanahan. 2006-03-09. Boundary slopes of 2-bridge links determine the crossing number. https://arxiv.org/abs/math/0603206

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