Geometry of alternating links on surfaces
HTML articles powered by AMS MathViewer
- by Joshua A. Howie and Jessica S. Purcell PDF
- Trans. Amer. Math. Soc. 373 (2020), 2349-2397 Request permission
Abstract:
We consider links that are alternating on surfaces embedded in a compact 3-manifold. We show that under mild restrictions, the complement of the link decomposes into simpler pieces, generalising the polyhedral decomposition of alternating links of Menasco. We use this to prove various facts about the hyperbolic geometry of generalisations of alternating links, including weakly generalised alternating links described by the first author. We give diagrammatical properties that determine when such links are hyperbolic, find the geometry of their checkerboard surfaces, bound volume, and exclude exceptional Dehn fillings.References
- C. Adams, C. Albors-Riera, B. Haddock, Z. Li, D. Nishida, B. Reinoso, and L. Wang, Hyperbolicity of links in thickened surfaces, Topology Appl. 256 (2019), 262–278. MR 3916014, DOI 10.1016/j.topol.2019.01.022
- C. Adams, A. Colestock, J. Fowler, W. Gillam, and E. Katerman, Cusp size bounds from singular surfaces in hyperbolic 3-manifolds, Trans. Amer. Math. Soc. 358 (2006), no. 2, 727–741. MR 2177038, DOI 10.1090/S0002-9947-05-03662-7
- Colin C. Adams, Toroidally alternating knots and links, Topology 33 (1994), no. 2, 353–369. MR 1273788, DOI 10.1016/0040-9383(94)90017-5
- Colin Adams, Hyperbolic knots, Handbook of knot theory, Elsevier B. V., Amsterdam, 2005, pp. 1–18. MR 2179259, DOI 10.1016/B978-044451452-3/50002-2
- Colin Adams, Noncompact Fuchsian and quasi-Fuchsian surfaces in hyperbolic 3-manifolds, Algebr. Geom. Topol. 7 (2007), 565–582. MR 2308957, DOI 10.2140/agt.2007.7.565
- Colin C. Adams, Jeffrey F. Brock, John Bugbee, Timothy D. Comar, Keith A. Faigin, Amy M. Huston, Anne M. Joseph, and David Pesikoff, Almost alternating links, Topology Appl. 46 (1992), no. 2, 151–165. MR 1184114, DOI 10.1016/0166-8641(92)90130-R
- Colin C. Adams, Aaron Calderon, and Nathaniel Mayer, Generalized bipyramids and hyperbolic volumes of alternating $k$-uniform tiling links, http://arxiv.org/abs/1709.00432, 2017.
- Ian Agol, Bounds on exceptional Dehn filling, Geom. Topol. 4 (2000), 431–449. MR 1799796, DOI 10.2140/gt.2000.4.431
- Ian Agol, Peter A. Storm, and William P. Thurston, Lower bounds on volumes of hyperbolic Haken 3-manifolds, J. Amer. Math. Soc. 20 (2007), no. 4, 1053–1077. With an appendix by Nathan Dunfield. MR 2328715, DOI 10.1090/S0894-0347-07-00564-4
- Robert J. Aumann, Asphericity of alternating knots, Ann. of Math. (2) 64 (1956), 374–392. MR 96236, DOI 10.2307/1969980
- Francis Bonahon, Bouts des variétés hyperboliques de dimension $3$, Ann. of Math. (2) 124 (1986), no. 1, 71–158 (French). MR 847953, DOI 10.2307/1971388
- Stephan D. Burton and Efstratia Kalfagianni, Geometric estimates from spanning surfaces, Bull. Lond. Math. Soc. 49 (2017), no. 4, 694–708. MR 3725490, DOI 10.1112/blms.12063
- R. D. Canary, D. B. A. Epstein, and P. Green, Notes on notes of Thurston, Analytical and geometric aspects of hyperbolic space (Coventry/Durham, 1984) London Math. Soc. Lecture Note Ser., vol. 111, Cambridge Univ. Press, Cambridge, 1987, pp. 3–92. MR 903850
- Chun Cao and G. Robert Meyerhoff, The orientable cusped hyperbolic $3$-manifolds of minimum volume, Invent. Math. 146 (2001), no. 3, 451–478. MR 1869847, DOI 10.1007/s002220100167
- Abhijit Champanerkar, Ilya Kofman, and Jessica S. Purcell, Geometrically and diagrammatically maximal knots, J. Lond. Math. Soc. (2) 94 (2016), no. 3, 883–908. MR 3614933, DOI 10.1112/jlms/jdw062
- Abhijit Champanerkar, Ilya Kofman, and Jessica S. Purcell, Geometry of biperiodic alternating links, J. Lond. Math. Soc. 99 (2019), no. 3, 807–830.
- Oliver T. Dasbach, David Futer, Efstratia Kalfagianni, Xiao-Song Lin, and Neal W. Stoltzfus, The Jones polynomial and graphs on surfaces, J. Combin. Theory Ser. B 98 (2008), no. 2, 384–399. MR 2389605, DOI 10.1016/j.jctb.2007.08.003
- Sérgio R. Fenley, Quasi-Fuchsian Seifert surfaces, Math. Z. 228 (1998), no. 2, 221–227. MR 1630563, DOI 10.1007/PL00004607
- David Futer and François Guéritaud, Angled decompositions of arborescent link complements, Proc. Lond. Math. Soc. (3) 98 (2009), no. 2, 325–364. MR 2481951, DOI 10.1112/plms/pdn033
- David Futer, Efstratia Kalfagianni, and Jessica S. Purcell, Dehn filling, volume, and the Jones polynomial, J. Differential Geom. 78 (2008), no. 3, 429–464. MR 2396249
- David Futer, Efstratia Kalfagianni, and Jessica Purcell, Guts of surfaces and the colored Jones polynomial, Lecture Notes in Mathematics, vol. 2069, Springer, Heidelberg, 2013. MR 3024600, DOI 10.1007/978-3-642-33302-6
- David Futer, Efstratia Kalfagianni, and Jessica S. Purcell, Quasifuchsian state surfaces, Trans. Amer. Math. Soc. 366 (2014), no. 8, 4323–4343. MR 3206461, DOI 10.1090/S0002-9947-2014-06182-5
- David Futer, Efstratia Kalfagianni, and Jessica S. Purcell, Hyperbolic semi-adequate links, Comm. Anal. Geom. 23 (2015), no. 5, 993–1030. MR 3458811, DOI 10.4310/CAG.2015.v23.n5.a3
- David Futer and Jessica S. Purcell, Links with no exceptional surgeries, Comment. Math. Helv. 82 (2007), no. 3, 629–664. MR 2314056, DOI 10.4171/CMH/105
- David Gabai, Quasi-minimal semi-Euclidean laminations in $3$-manifolds, Surveys in differential geometry, Vol. III (Cambridge, MA, 1996) Int. Press, Boston, MA, 1998, pp. 195–242. MR 1677889
- Joshua Evan Greene, Alternating links and definite surfaces, Duke Math. J. 166 (2017), no. 11, 2133–2151. With an appendix by András Juhász and Marc Lackenby. MR 3694566, DOI 10.1215/00127094-2017-0004
- Chuichiro Hayashi, Links with alternating diagrams on closed surfaces of positive genus, Math. Proc. Cambridge Philos. Soc. 117 (1995), no. 1, 113–128. MR 1297898, DOI 10.1017/S0305004100072947
- Joshua A. Howie, Surface-alternating knots and links, Ph.D. thesis, University of Melbourne, 2015.
- Joshua A. Howie, A characterisation of alternating knot exteriors, Geom. Topol. 21 (2017), no. 4, 2353–2371. MR 3654110, DOI 10.2140/gt.2017.21.2353
- Joshua A. Howie and J. Hyam Rubinstein, Weakly generalised alternating links, in preparation, 2019.
- Kazuhiro Ichihara and Hidetoshi Masai, Exceptional surgeries on alternating knots, Comm. Anal. Geom. 24 (2016), no. 2, 337–377. MR 3514563, DOI 10.4310/CAG.2016.v24.n2.a5
- Marc Lackenby, Word hyperbolic Dehn surgery, Invent. Math. 140 (2000), no. 2, 243–282. MR 1756996, DOI 10.1007/s002220000047
- Marc Lackenby, The volume of hyperbolic alternating link complements, Proc. London Math. Soc. (3) 88 (2004), no. 1, 204–224. With an appendix by Ian Agol and Dylan Thurston. MR 2018964, DOI 10.1112/S0024611503014291
- Marc Lackenby and Jessica S. Purcell, Cusp volumes of alternating knots, Geom. Topol. 20 (2016), no. 4, 2053–2078. MR 3548463, DOI 10.2140/gt.2016.20.2053
- William W. Menasco, Polyhedra representation of link complements, Low-dimensional topology (San Francisco, Calif., 1981) Contemp. Math., vol. 20, Amer. Math. Soc., Providence, RI, 1983, pp. 305–325. MR 718149, DOI 10.1090/conm/020/718149
- W. Menasco, Closed incompressible surfaces in alternating knot and link complements, Topology 23 (1984), no. 1, 37–44. MR 721450, DOI 10.1016/0040-9383(84)90023-5
- Louise Moser, Elementary surgery along a torus knot, Pacific J. Math. 38 (1971), 737–745. MR 383406, DOI 10.2140/pjm.1971.38.737
- Makoto Ozawa, Non-triviality of generalized alternating knots, J. Knot Theory Ramifications 15 (2006), no. 3, 351–360. MR 2217501, DOI 10.1142/S0218216506004506
- Makoto Ozawa, Essential state surfaces for knots and links, J. Aust. Math. Soc. 91 (2011), no. 3, 391–404. MR 2900614, DOI 10.1017/S1446788712000055
- Makoto Ozawa and Joachim Hyam Rubinstein, On the Neuwirth conjecture for knots, Comm. Anal. Geom. 20 (2012), no. 5, 1019–1060. MR 3053620, DOI 10.4310/CAG.2012.v20.n5.a5
- Makoto Ozawa and Yukihiro Tsutsumi, Totally knotted Seifert surfaces with accidental peripherals, Proc. Amer. Math. Soc. 131 (2003), no. 12, 3945–3954. MR 1999945, DOI 10.1090/S0002-9939-03-06964-8
- Morwen Thistlethwaite and Anastasiia Tsvietkova, An alternative approach to hyperbolic structures on link complements, Algebr. Geom. Topol. 14 (2014), no. 3, 1307–1337. MR 3190595, DOI 10.2140/agt.2014.14.1307
- William Thurston, The geometry and topology of three-manifolds, http://www.msri.org/gt3m/, 1979.
- William P. Thurston, Three-dimensional manifolds, Kleinian groups and hyperbolic geometry, Bull. Amer. Math. Soc. (N.S.) 6 (1982), no. 3, 357–381. MR 648524, DOI 10.1090/S0273-0979-1982-15003-0
Additional Information
- Joshua A. Howie
- Affiliation: School of Mathematical Sciences, Monash University, Victoria 3800, Australia
- Address at time of publication: Department of Mathematics, University of California, Davis, Davis, California 95616
- Email: josh.howie@monash.edu
- Jessica S. Purcell
- Affiliation: School of Mathematical Sciences, Monash University, Victoria 3800, Australia
- MR Author ID: 807518
- ORCID: 0000-0002-0618-2840
- Email: jessica.purcell@monash.edu
- Received by editor(s): December 6, 2018
- Received by editor(s) in revised form: June 5, 2019
- Published electronically: January 23, 2020
- Additional Notes: The first author was supported by a Lift-off Fellowship from the Australian Mathematical Society.
Both authors were partially supported by grants from the Australian Research Council. - © Copyright 2020 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 373 (2020), 2349-2397
- MSC (2010): Primary 57M27, 57M20
- DOI: https://doi.org/10.1090/tran/7929
- MathSciNet review: 4069222