Download this article
 Download this article For screen
For printing
Recent Issues

Volume 30
Issue 9, 3169–3578
Issue 8, 2779–3167
Issue 7, 2431–2778
Issue 6, 2043–2429
Issue 5, 1611–2042
Issue 4, 1203–1610
Issue 3, 835–1201
Issue 2, 389–833
Issue 1, 1–388

Volume 29, 9 issues

Volume 28, 9 issues

Volume 27, 9 issues

Volume 26, 8 issues

Volume 25, 7 issues

Volume 24, 7 issues

Volume 23, 7 issues

Volume 22, 7 issues

Volume 21, 6 issues

Volume 20, 6 issues

Volume 19, 6 issues

Volume 18, 5 issues

Volume 17, 5 issues

Volume 16, 4 issues

Volume 15, 4 issues

Volume 14, 5 issues

Volume 13, 5 issues

Volume 12, 5 issues

Volume 11, 4 issues

Volume 10, 4 issues

Volume 9, 4 issues

Volume 8, 3 issues

Volume 7, 2 issues

Volume 6, 2 issues

Volume 5, 2 issues

Volume 4, 1 issue

Volume 3, 1 issue

Volume 2, 1 issue

Volume 1, 1 issue

The Journal
About the journal
Ethics and policies
Peer-review process
 
Submission guidelines
Submission form
Editorial board
 
Subscriptions
 
ISSN (electronic): 1364-0380
ISSN (print): 1465-3060
 
Author index
To appear
 
Other MSP journals
Knot complements decomposing into prisms

Jason DeBlois, Arshia Gharagozlou and Neil R Hoffman

Geometry & Topology 30 (2026) 3361–3422
Bibliography
1 C C Adams, Toroidally alternating knots and links, Topology 33 (1994) 353 MR1273788
2 C Adams, Triple crossing number of knots and links, J. Knot Theory Ramifications 22 (2013) 1350006 MR3037297
3 C Adams, E Schoenfeld, Totally geodesic Seifert surfaces in hyperbolic knot and link complements, I, Geom. Dedicata 116 (2005) 237 MR2195448
4 C C Adams, J F Brock, J Bugbee, T D Comar, K A Faigin, A M Huston, A M Joseph, D Pesikoff, Almost alternating links, Topology Appl. 46 (1992) 151 MR1184114
5 I Agol, Bounds on exceptional Dehn filling, Geom. Topol. 4 (2000) 431 MR1799796
6 I R Aitchison, J H Rubinstein, Combinatorial cubings, cusps, and the dodecahedral knots, from: "Topology ’90" (editors B Apanasov, W D Neumann, A W Reid, L Siebenmann), Ohio State Univ. Math. Res. Inst. Publ. 1, de Gruyter (1992) 17 MR1184399
7 I R Aitchison, J H Rubinstein, Geodesic surfaces in knot complements, Experiment. Math. 6 (1997) 137 MR1474574
8 E M Andreev, Convex polyhedra of finite volume in Lobačevskiĭ space, Mat. Sb. (N.S.) 83(125) (1970) 256 MR273510
9 U Bader, D Fisher, N Miller, M Stover, Arithmeticity, superrigidity, and totally geodesic submanifolds, Ann. of Math. (2) 193 (2021) 837 MR4250391
10 M Boileau, S Boyer, R Cebanu, G S Walsh, Knot commensurability and the Berge conjecture, Geom. Topol. 16 (2012) 625 MR2928979
11 M Boileau, S Boyer, R Cebanu, G S Walsh, Knot complements, hidden symmetries and reflection orbifolds, Ann. Fac. Sci. Toulouse Math. (6) 24 (2015) 1179 MR3485331
12 B A Burton, Detecting genus in vertex links for the fast enumeration of 3-manifold triangulations, from: "ISSAC 2011—Proceedings of the 36th International Symposium on Symbolic and Algebraic Computation" (editor A Leykin), ACM (2011) 59 MR2895195
13 B A Burton, R Budney, W Pettersson, Regina: software for low-dimensional topology, software (1999–2025)
14 D Calegari, Real places and torus bundles, Geom. Dedicata 118 (2006) 209 MR2239457
15 E Chesebro, J DeBlois, N R Hoffman, C Millichap, P Mondal, W Worden, Dehn surgery and hyperbolic knot complements without hidden symmetries, Int. Math. Res. Not. 2023 (2023) 5293 MR4565689
16 E Chesebro, J DeBlois, P Mondal, Generic hyperbolic knot complements without hidden symmetries, Comm. Anal. Geom. 32 (2024) 255 MR4806767
17 M Culler, N Dunfield, M Goerner, The low_index project, a computer program implementing the Sims algorithm, version 1.1 (2022)
18 M Culler, N Dunfield, M Goerner, J Weeks, SnapPy, a computer program for studying the geometry and topology of 3-manifolds, software (2024)
19 J DeBlois, A Gharagozlou, N R Hoffman, Knot complements decomposing into prisms, (2025) arXiv:2507.01263
20 W D Dunbar, G R Meyerhoff, Volumes of hyperbolic 3-orbifolds, Indiana Univ. Math. J. 43 (1994) 611 MR1291531
21 T GAP Group, GAP—Groups, Algorithms, and Programming, version 4.11.1 (2021)
22 A Gharagozlou, Enumerating manifold covers of prism orbifolds, PhD thesis, University of Pittsburgh (2025) MR4916358
23 A Hagberg, D Schult, P Swart, Exploring network structure, dynamics, and function using networkx, from: "Proceedings of the 7th Python in Science Conference" (editors G Varoquaux, T Vaught, J Millman) (2008) 11
24 N Hoffman, Commensurability classes containing three knot complements, Algebr. Geom. Topol. 10 (2010) 663 MR2606796
25 N R Hoffman, On knot complements that decompose into regular ideal dodecahedra, Geom. Dedicata 173 (2014) 299 MR3275305
26 N R Hoffman, Small knot complements, exceptional surgeries and hidden symmetries, Algebr. Geom. Topol. 14 (2014) 3227 MR3302960
27 N R Hoffman, Cusp types of quotients of hyperbolic knot complements, Proc. Amer. Math. Soc. Ser. B 9 (2022) 336 MR4470781
28 N R Hoffman, C Millichap, W Worden, Symmetries and hidden symmetries of (𝜖,dL)-twisted knot complements, Algebr. Geom. Topol. 22 (2022) 601 MR4464461
29 T Homma, On the existence of unknotted polygons on 2-manifolds in E3, Osaka Math. J. 6 (1954) 129 MR63672
30 K Ichihara, M Ozawa, Hyperbolic knot complements without closed embedded totally geodesic surfaces, J. Austral. Math. Soc. Ser. A 68 (2000) 379 MR1753367
31 W Jaco, J H Rubinstein, 0-efficient triangulations of 3-manifolds, J. Differential Geom. 65 (2003) 61 MR2057531
32 R Kirby, Problems in low-dimensional topology, from: "Geometric topology" (editor R Kirby), AMS/IP Stud. Adv. Math. 2.2, Amer. Math. Soc. (1997) 35 MR1470751
33 M Lackenby, Word hyperbolic Dehn surgery, Invent. Math. 140 (2000) 243 MR1756996
34 G S Lakeland, C G Roth, Hyperbolic triangular prisms with one ideal vertex, Involve 13 (2020) 361 MR4129388
35 K Le, R Palmer, Totally geodesic surfaces in twist knot complements, Pacific J. Math. 319 (2022) 153 MR4475678
36 K Le, R Palmer, Geodesic surfaces in the complement of knots with small crossing number, New York J. Math. 29 (2023) 363 MR4564012
37 C J Leininger, Small curvature surfaces in hyperbolic 3-manifolds, J. Knot Theory Ramifications 15 (2006) 379 MR2217503
38 M T Lozano, J H Przytycki, Incompressible surfaces in the exterior of a closed 3-braid, I : Surfaces with horizontal boundary components, Math. Proc. Cambridge Philos. Soc. 98 (1985) 275 MR795894
39 M L Macasieb, T W Mattman, Commensurability classes of (−2,3,n) pretzel knot complements, Algebr. Geom. Topol. 8 (2008) 1833 MR2448875
40 C Maclachlan, A W Reid, The arithmetic of hyperbolic 3-manifolds, 219, Springer (2003) MR1937957
41 B Maskit, On Poincaré’s theorem for fundamental polygons, Advances in Math. 7 (1971) 219 MR297997
42 H Matsuda, Complements of hyperbolic knots of braid index four contain no closed embedded totally geodesic surfaces, Topology Appl. 119 (2002) 1 MR1881706
43 W Menasco, A W Reid, Totally geodesic surfaces in hyperbolic link complements, from: "Topology ’90" (editors B Apanasov, W D Neumann, A W Reid, L Siebenmann), Ohio State Univ. Math. Res. Inst. Publ. 1, de Gruyter (1992) 215 MR1184413
44 C Millichap, Mutations and short geodesics in hyperbolic 3-manifolds, Comm. Anal. Geom. 25 (2017) 625 MR3702548
45 W Narkiewicz, Elementary and analytic theory of algebraic numbers, 57, PWN (1974) MR347767
46 W D Neumann, A W Reid, Arithmetic of hyperbolic manifolds, from: "Topology ’90" (editors B Apanasov, W D Neumann, A W Reid, L Siebenmann), Ohio State Univ. Math. Res. Inst. Publ. 1, de Gruyter (1992) 273 MR1184416
47 U Oertel, Closed incompressible surfaces in complements of star links, Pacific J. Math. 111 (1984) 209 MR732067
48 M Ozawa, Unknotting submanifolds of the 3-sphere by twistings, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 18 (2018) 1145 MR3807598
49 J G Ratcliffe, Foundations of hyperbolic manifolds, 149, Springer (2019) MR4221225
50 A W Reid, Arithmeticity of knot complements, J. London Math. Soc. (2) 43 (1991) 171 MR1099096
51 A W Reid, G S Walsh, Commensurability classes of 2-bridge knot complements, Algebr. Geom. Topol. 8 (2008) 1031 MR2443107
52 R F Riley, Parabolic representations and symmetries of the knot 932, from: "Computers in geometry and topology" (editor M C Tangora), Lecture Notes in Pure and Appl. Math. 114, Dekker (1989) 297 MR988702
53 D Ruberman, Mutation and volumes of knots in S3, Invent. Math. 90 (1987) 189 MR906585
54 T S developers, SageMath, the Sage Mathematics Software System, version 10.0 (2023)
55 W P Thurston, The geometry and topology of three-manifolds, lecture notes (1979)