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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: [email protected]
STEVEN SIVEK
Affiliation:
Department of Mathematics, Princeton University, Fine Hall, Washington Road, Princeton, NJ 08544-1000, U.S.A. e-mail: [email protected]
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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.

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 

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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