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We count the number of isotopy classes of closed, connected, orientable,essential surfaces embedded in the exterior B of the knot K13n586.The mainresult is that the count of surfaces by genus is equal to the Euler totentfunction. This is the first manifold for which we know the number of surfacesfor any genus. The main argument is to show when normal surfaces in B areconnected by counting their number of components. We implement tools from Agol,Hass and Thurston to convert the problem of counting components of surfacesinto counting the number of orbits in a set of integers under a collection ofbijections defined on its subsets.

Essential Surfaces in the Exterior of K13n586