Hostname: page-component-7f64f4797f-l842n Total loading time: 0 Render date: 2025-11-04T02:36:17.162Z Has data issue: false hasContentIssue false
  • English
  • Français

Quasi-alternating links with small determinant

Published online by Cambridge University Press:  14 July 2016

TYE LIDMAN  and
STEVEN SIVEK
TYE LIDMAN
Affiliation:
Department of Mathematics, Box 8205, NC State University, Raleigh, NC 27695-8205, U.S.A. e-mail: tlidman@ncsu.edu
STEVEN SIVEK
Affiliation:
Department of Mathematics, Princeton University, Fine Hall, Washington Road, Princeton, NJ 08544-1000, U.S.A. e-mail: ssivek@math.princeton.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

Quasi-alternating links of determinant 1, 2, 3 and 5 were previously classified by Greene and Teragaito, who showed that the only such links are two-bridge. In this paper, we extend this result by showing that all quasi-alternating links of determinant at most 7 are connected sums of two-bridge links, which is optimal since there are quasi-alternating links not of this form for all larger determinants. We achieve this by studying their branched double covers and characterising distance-one surgeries between lens spaces of small order, leading to a classification of formal L-spaces with order at most 7.

Information

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 162 , Issue 2 , March 2017 , pp. 319 - 336
Copyright
Copyright © Cambridge Philosophical Society 2016 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

REFERENCES

[BL90] Boyer, S. and Lines, D. Surgery formulae for Casson's invariant and extensions to homology lens spaces. J. Reine Angew. Math. 405 (1990), 181–220.Google Scholar
[CGO01] Cochran, T. D., Gerges, A. and Orr, K. Dehn surgery equivalence relations on 3-manifolds. Math. Proc. Camb. Phil. Soc. 131 (1) (2001), 97–127.Google Scholar
[CK09] Champanerkar, A. and Kofman, I. Twisting quasi-alternating links. Proc. Amer. Math. Soc. 137 (7) (2009), 2451–2458.Google Scholar
[Gai14] Gainullin, F. Only finitely many alternating knots can yield a given manifold by surgery. arXiv 1411.2249 (2014).Google Scholar
[Gai15] Gainullin, F. Heegaard Floer homology and knots determined by their complements. arXiv:1504.06180 (2015).Google Scholar
[GL14] Greene, J. E. and Levine, A. S. Strong Heegaard diagrams and strong L-spaces. arXiv:1411.6329 (2014).Google Scholar
[Gre10] Greene, J. Homologically thin, non-quasi-alternating links. Math. Res. Lett. 17 (1) (2010), 39–49.Google Scholar
[Gre15] Greene, J. E. L-space surgeries, genus bounds, and the cabling conjecture. J. Differential Geom. 100 (3) (2015), 491–506.Google Scholar
[HR85] Hodgson, C. and Rubinstein, J. H. Involutions and isotopies of lens spaces. In Knot theory and manifolds (Vancouver, B.C., 1983) Lecture Notes in Math. vol. 1144 (Springer, Berlin, 1985) pp. 60–96.Google Scholar
[JK13] Jong, I. D. and Kishimoto, K. On positive knots of genus two. Kobe J. Math. 30 (1-2) (2013), 1–18.Google Scholar
[KM07] Kronheimer, P. and Mrowka, T. Monopoles and three-manifolds New Mathematical Monographs, vol. 10 (Cambridge University Press, Cambridge, 2007).Google Scholar
[KMOS07] Kronheimer, P., Mrowka, T., Ozsváth, P. and Szabó, Z.. Monopoles and lens space surgeries. Ann. of Math. (2) 165 (2) (2007), 457–546.CrossRefGoogle Scholar
[KT80] Kim, P. K. and Tollefson, J. L. Splitting the PL involutions of nonprime 3-manifolds. Michigan Math. J. 27 (3) (1980), 259–274.Google Scholar
[LP14] Lackenby, M. and Purcell, J. S. Cusp volumes of alternating knots. arXiv 1410.6297 (2014).Google Scholar
[MO08] Manolescu, C. and Ozsváth, P. On the Khovanov and knot Floer homologies of quasi-alternating links. In Proceedings of Gökova Geometry-Topology Conference 2007, (Gökova Geometry/Topology Conference (GGT), Gökova, 2008), pp. 60–81.Google Scholar
[Mos71] Moser, L. Elementary surgery along a torus knot. Pacific J. Math. 38 (1971), 737–745.Google Scholar
[Ni06] Ni, Y. A note on knot Floer homology of links. Geom. Topol. 10 (2006), 695–713.Google Scholar
[OS03] Ozsváth, P. and Szabó, Z. Absolutely graded Floer homologies and intersection forms for four-manifolds with boundary. Adv. Math. 173 (2) (2003), 179–261.Google Scholar
[OS04] Ozsváth, P. and Szabó, Z. Holomorphic disks and three-manifold invariants: properties and applications. Ann. of Math. (2) 159 (3) (2004), 1159–1245.CrossRefGoogle Scholar
[OS05] Ozsváth, P. and Szabó, Z. On the Heegaard Floer homology of branched double-covers. Adv. Math. 194 (1) (2005), 1–33.Google Scholar
[OS06] Ozsváth, P. and Szabó, Z. Holomorphic triangles and invariants for smooth four-manifolds. Adv. Math. 202 (2) (2006), 326–400.Google Scholar
[OS11] Ozsváth, P. S. and Szabó, Z. Knot Floer homology and rational surgeries. Algebr. Geom. Topol. 11 (1) (2011), 1–68.Google Scholar
[RG72] Rademacher, H. and Grosswald, E. Dedekind sums. The Mathematical Association of America (Washington, D.C., 1972). The Carus Mathematical Monographs, No. 16.Google Scholar
[Rus04] Rustamov, R. Surgery formula for the renormalised Euler characteristic of Heegaard Floer homology. arXiv:math.GT/0409294 (2004).Google Scholar
[Tan09] Tange, M. Ozsváth Szabó's correction term and lens surgery. Math. Proc. Camb. Phil. Soc. 146 (1) (2009), 119–134.Google Scholar
[Ter14] Teragaito, M. Quasi-alternating links and Q-polynomials. J. Knot Theory Ramifications 23 (12) (2014), 1450068, 6.Google Scholar
[Ter15] Teragaito, M. Quasi-alternating links and Kauffman polynomials. J. Knot Theory Ramifications 24 (7) (2015), 1550038 (17 pages).Google Scholar
[Wal92] Walker, K. An extension of Casson's invariant Annals of Mathematics Studies. vol. 126 (Princeton University Press, Princeton, NJ, 1992).Google Scholar