Browse > Article
http://dx.doi.org/10.4134/BKMS.b210503

KNOTS IN HOMOLOGY LENS SPACES DETERMINED BY THEIR COMPLEMENTS  

Ichihara, Kazuhiro (Department of Mathematics College of Humanities and Sciences Nihon University)
Saito, Toshio (Department of Mathematics Joetsu University of Education)
Publication Information
Bulletin of the Korean Mathematical Society / v.59, no.4, 2022 , pp. 869-877 More about this Journal
Abstract
In this paper, we consider the knot complement problem for not null-homologous knots in homology lens spaces. Let M be a homology lens space with H1(M; ℤ) ≅ ℤp and K a not null-homologous knot in M. We show that, K is determined by its complement if M is non-hyperbolic, K is hyperbolic, and p is a prime greater than 7, or, if M is actually a lens space L(p, q) and K represents a generator of H1(L(p, q)).
Keywords
Knot complement; homology lens space;
Citations & Related Records
연도 인용수 순위
  • Reference
1 Y. Mathieu, Closed 3-manifolds unchanged by Dehn surgery, J. Knot Theory Ramifications 1 (1992), no. 3, 279-296. https://doi.org/10.1142/S0218216592000161   DOI
2 P. Ozsvath and Z. Szabo, Lectures on Heegaard Floer homology, in Floer homology, gauge theory, and low-dimensional topology, 29-70, Clay Math. Proc., 5, Amer. Math. Soc., Providence, RI, 2006.
3 S. A. Bleiler, C. D. Hodgson, and J. R. Weeks, Cosmetic surgery on knots, in Proceedings of the Kirbyfest (Berkeley, CA, 1998), 23-34, Geom. Topol. Monogr., 2, Geom. Topol. Publ., Coventry, 1999. https://doi.org/10.2140/gtm.1999.2.23
4 A. Christensen, Homology of manifolds obtained by Dehn surgery on knots in lens spaces, J. Knot Theory Ramifications 9 (2000), no. 4, 431-442. https://doi.org/10.1142/S0218216500000219   DOI
5 M. Culler, C. M. Gordon, J. Luecke, and P. B. Shalen, Dehn surgery on knots, Ann. of Math. (2) 125 (1987), no. 2, 237-300. https://doi.org/10.2307/1971311   DOI
6 K. Ichihara and I. D. Jong, Cosmetic banding on knots and links, Osaka J. Math. 55 (2018), no. 4, 731-745. https://projecteuclid.org/euclid.ojm/1539158668
7 M. Lackenby and R. Meyerhoff, The maximal number of exceptional Dehn surgeries, Invent. Math. 191 (2013), no. 2, 341-382. https://doi.org/10.1007/s00222-012-0395-2   DOI
8 F. Gainullin, Heegaard Floer homology and knots determined by their complements, Algebr. Geom. Topol. 18 (2018), no. 1, 69-109. https://doi.org/10.2140/agt.2018.18.69   DOI
9 Y. W. Rong, Some knots not determined by their complements, in Quantum topology, 339-353, Ser. Knots Everything, 3, World Sci. Publ., River Edge, NJ, 1993. https://doi.org/10.1142/9789812796387_0019   DOI
10 S. Boyer and X. Zhang, Finite Dehn surgery on knots, J. Amer. Math. Soc. 9 (1996), no. 4, 1005-1050. https://doi.org/10.1090/S0894-0347-96-00201-9   DOI
11 C. McA. Gordon, Dehn surgery on knots, in Proceedings of the International Congress of Mathematicians, Vol. I, II (Kyoto, 1990), 631-642, Math. Soc. Japan, Tokyo, 1991.
12 C. McA. Gordon and J. Luecke, Knots are determined by their complements, J. Amer. Math. Soc. 2 (1989), no. 2, 371-415. https://doi.org/10.2307/1990979   DOI
13 E. Luft and D. Sjerve, Degree-1 maps into lens spaces and free cyclic actions on homology 3-spheres, Topology Appl. 37 (1990), no. 2, 131-136. https://doi.org/10.1016/0166-8641(90)90057-9   DOI
14 C. R. Guilbault, Homology lens spaces and Dehn surgery on homology spheres, Fund. Math. 144 (1994), no. 3, 287-292.   DOI
15 T. Ito, Applications of the Casson-Walker invariant to the knot complement and the cosmetic crossing conjectures, preprint, arXiv:2103.15277.
16 R. Kirby, Problems in low-dimensional topology, in Geometric topology (Athens, GA, 1993), 35-473, AMS/IP Stud. Adv. Math., 2.2, Amer. Math. Soc., Providence, RI, 1997.
17 Y. Mathieu, Sur les noeuds qui ne sont pas determines par leur complement et problemes de chirurgie dans les varietes de dimension 3, These, L'Universite de Provence, 1990.
18 D. Matignon, On the knot complement problem for non-hyperbolic knots, Topology Appl. 157 (2010), no. 12, 1900-1925. https://doi.org/10.1016/j.topol.2010.03.009   DOI
19 H. Tietze, Uber die topologischen Invarianten mehrdimensionaler Mannigfaltigkeiten, Monatsh. Math. Phys. 19 (1908), no. 1, 1-118. https://doi.org/10.1007/BF01736688   DOI