Essential Surfaces in the Exterior of K13n586

2021/10/19 by Chaeryn Lee, Lee, Chaeryn
Mathematics · #57M50 #Advanced Combinatorial Mathematics #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Mathematical Dynamics and Fractals

paper · pdf · doi:10.48550/arxiv.2110.10231

openalex publication_date 2021/10/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 totent 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.

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