Abstract: We count the number of isotopy classes of closed, connected, orientable, essential surfaces embedded in the exterior B of the knot K13n586. The main result is that the count of surfaces by genus is equal to the Euler totient function. This is the first manifold for which we know the number of surfaces for any genus. The main argument is to show when normal surfaces in B are connected by counting their number of components. We implement tools from Agol, Hass and Thurston to convert the problem of counting components of surfaces into counting the number of orbits in a set of integers under a collection of bijections defined on its subsets.
In this talk we will focus on understanding these techniques from Agol, Hass and Thurston and how they are applied to the specific case of our knot exterior.
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