Fully augmented links in the thickened torus

Kwon, Alice¹

¹Science Department, SUNY Maritime, Throggs Neck, NY, United States

Algebraic and Geometric Topology, Tome 25 (2025) no. 3, pp. 1411-1432

Résumé

We study the geometry of fully augmented link complements in the thickened torus and describe their geometric properties, generalizing the study of fully augmented links in S3. We classify which fully augmented links in the thickened torus are hyperbolic, and show that their complements in the thickened torus decompose into ideal right-angled torihedra. We also study volume density of fully augmented links in S3, defined to be the ratio of its volume and the number of augmentations. We prove the volume density conjecture for fully augmented links, which states that the volume density of a sequence of fully augmented links in S3 which diagrammatically converges to a biperiodic link converges to the volume density of that biperiodic link. Furthermore, we show that the complement of a sequence of these links approaches the complement of the biperiodic link as a geometric limit.

Détail

DOI : 10.2140/agt.2025.25.1411

Keywords: geometric topology, fully augmented links

Bibliographie

[1] C C Adams, Thrice-punctured spheres in hyperbolic 3–manifolds, Trans. Amer. Math. Soc. 287 (1985) 645 | DOI

[2] C C Adams, Augmented alternating link complements are hyperbolic, from: "Low-dimensional topology and Kleinian groups", Lond. Math. Soc. Lect. Note Ser. 112, Cambridge Univ. Press (1986) 115

[3] C Adams, A Calderon, N Mayer, Generalized bipyramids and hyperbolic volumes of alternating k–uniform tiling links, Topology Appl. 271 (2020) 107045 | DOI

[4] C Adams, M Capovilla-Searle, D Li, L Q Li, J Mcerlean, A Simons, N Stewart, X Wang, Augmented cellular alternating links in thickened surfaces are hyperbolic, Eur. J. Math. 9 (2023) 100 | DOI

[5] C K Atkinson, Volume estimates for equiangular hyperbolic Coxeter polyhedra, Algebr. Geom. Topol. 9 (2009) 1225 | DOI

[6] L Babai, The growth rate of vertex-transitive planar graphs, from: "Proceedings of the eighth annual ACM–SIAM symposium on discrete algorithms", ACM (1997) 564

[7] A I Bobenko, B A Springborn, Variational principles for circle patterns and Koebe’s theorem, Trans. Amer. Math. Soc. 356 (2004) 659 | DOI

[8] A Champanerkar, D Futer, I Kofman, W Neumann, J S Purcell, Volume bounds for generalized twisted torus links, Math. Res. Lett. 18 (2011) 1097 | DOI

[9] A Champanerkar, I Kofman, Determinant density and biperiodic alternating links, New York J. Math. 22 (2016) 891

[10] A Champanerkar, I Kofman, J S Purcell, Geometrically and diagrammatically maximal knots, J. Lond. Math. Soc. 94 (2016) 883 | DOI

[11] A Champanerkar, I Kofman, J S Purcell, Geometry of biperiodic alternating links, J. Lond. Math. Soc. 99 (2019) 807 | DOI

[12] E Chesebro, J Deblois, H Wilton, Some virtually special hyperbolic 3–manifold groups, Comment. Math. Helv. 87 (2012) 727 | DOI

[13] J A Howie, J S Purcell, Geometry of alternating links on surfaces, Trans. Amer. Math. Soc. 373 (2020) 2349 | DOI

[14] A Kwon, Y H Tham, Hyperbolicity of augmented links in the thickened torus, J. Knot Theory Ramifications 31 (2022) 2250025 | DOI

[15] M Lackenby, The volume of hyperbolic alternating link complements, Proc. Lond. Math. Soc. 88 (2004) 204 | DOI

[16] Y Miyamoto, Volumes of hyperbolic manifolds with geodesic boundary, Topology 33 (1994) 613 | DOI

[17] J S Purcell, An introduction to fully augmented links, from: "Interactions between hyperbolic geometry, quantum topology and number theory", Contemp. Math. 541, Amer. Math. Soc. (2011) 205 | DOI

[18] J S Purcell, Hyperbolic knot theory, 209, Amer. Math. Soc. (2020) | DOI

[19] W P Thurston, The geometry and topology of three-manifolds, lecture notes (1979)