2022/04/04 by Brannon Basilio, Basilio, Brannon
Mathematics · #57K10 (Primary) #57K32 (Secondary) #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology
paper · pdf · doi:10.48550/arxiv.2204.01789
openalex publication_date 2022/04/04 · openalex created_date 2022/04/26 · openalex updated_date 2026/07/28
In the complement of a hyperbolic Montesinos knot with 4 rational tangles, we investigate the number of closed, connected, essential, orientable surfaces of a fixed genus g, up to isotopy. We show that there are exactly 12 genus 2 surfaces and 8ϕ(g - 1) surfaces of genus greater than 2, where ϕ(g - 1) is the Euler totient function of g - 1. Observe that this count is independent of the number of crossings of the knot. Moreover, this class of knots form an infinite class of hyperbolic 3-manifolds and the result applies to all such knot complements.