2024/03/19 by Basilio, Brannon, Lee, Chaeryn, Malionek, Joseph · 1 citation
#57K10 (Secondary) #57K32 (Primary) #FOS: Mathematics #G.4 #Geometric Topology (math.GT)
paper · doi:10.48550/arxiv.2403.12397
Finding a totally geodesic surface, an embedded surface where the geodesics in the surface are also geodesics in the surrounding manifold, has been a problem of interest in the study of 3-manifolds. This has especially been of interest in hyperbolic 3-manifolds and knot complements, complements of piecewise-linearly embedded circles in the 3-sphere. This is due to Menasco-Reid's conjecture stating that hyperbolic knot complements do not contain such surfaces. Here, we present an algorithm that determines whether a given surface is totally geodesic and an algorithm that checks whether a given 3-manifold contains a totally geodesic surface. We applied our algorithm on over 150,000 3-manifolds and discovered nine 3-manifolds with totally geodesic surfaces. Additionally, we verified Menasco-Reid's conjecture for knots up to 12 crossings.