Special Issue for Tim Cochran; Guest Editors: J. E. Grigsby, S. Harvey, K. Orr and D. RubermanNo Access

A note on applications of the $d$ $d$ -invariant and Donaldson's theorem

Joshua Evan Greene

Department of Mathematics, Boston College Chestnut Hill, MA 02467, USA

https://doi.org/10.1142/S0218216517400065Cited by:1 (Source: Crossref)

Abstract

This paper contains two remarks about the application of the $d$ $d$ -invariant in Heegaard Floer homology and Donaldson's diagonalization theorem to knot theory. The first is the equivalence of two obstructions they give to a 2-bridge knot being smoothly slice. The second carries out a suggestion by Stefan Friedl to replace the use of Heegaard Floer homology by Donaldson's theorem in the proof of the main result of [J. E. Greene, Lattices, graphs, and Conway mutation, Invent. Math.192(3) (2013) 717–750] concerning Conway mutation of alternating links.

Keywords:

AMSC: 05C21, 05C50, 11H55, 57M15, 57M25, 57M27

References

1. A. J. Casson and C. McA. Gordon, Cobordism of classical knots, in À la Recherche de la Topologie Perdue, Progress in Mathematics, Vol. 62 (Birkhäuser, Boston, MA, 1986), pp. 181–199. Google Scholar
2. T. D. Cochran and R. E. Gompf, Applications of Donaldson's theorems to classical knot concordance, homology $3$ $3$ -spheres and property $P$ $P$ , Topology 27(4) (1988) 495–512. Crossref, Web of Science, Google Scholar
3. S. K. Donaldson, The orientation of Yang-Mills moduli spaces and $4$ $4$ -manifold topology, J. Differential Geom. 26(3) (1987) 397–428. Crossref, Web of Science, Google Scholar
4. N. D. Elkies, A characterization of the $ℤ^{n}$ lattice, Math. Res. Lett. 2(3) (1995) 321–326. Crossref, Web of Science, Google Scholar
5. M. H. Freedman and F. Quinn, Topology of 4-Manifolds, Princeton Mathematical Series, Vol. 39 (Princeton University Press, Princeton, NJ, 1990). Google Scholar
6. R. E. Gompf and A. I. Stipsicz, 4-Manifolds and Kirby Calculus, Graduate Studies in Mathematics, Vol. 20 (American Mathematical Society, Providence, RI, 1999). Crossref, Google Scholar
7. C. McA. Gordon and R. A. Litherland, On the signature of a link, Invent. Math. 47(1) (1978) 53–69. Crossref, Web of Science, Google Scholar
8. J. E. Greene, Conway mutation and alternating links, in Proce. Gökova Geometry-Topology Conf. 2011 (International Press, Somerville, MA, 2012), pp. 31–41. Google Scholar
9. J. E. Greene, Lattices, graphs, and Conway mutation, Invent. Math. 192(3) (2013) 717–750. Crossref, Web of Science, Google Scholar
10. J. E. Greene and S. Jabuka, The slice-ribbon conjecture for $3$ -stranded pretzel knots, Amer. J. Math. 133(3) (2011) 555–580. Crossref, Web of Science, Google Scholar
11. J. E. Grigsby, D. Ruberman and S. Strle, Knot concordance and Heegaard Floer homology invariants in branched covers, Geom. Topol. 12(4) (2008) 2249–2275. Crossref, Web of Science, Google Scholar
12. S. Jabuka and S. Naik, Order in the concordance group and Heegaard Floer homology, Geom. Topol. 11 (2007) 979–994. Crossref, Web of Science, Google Scholar
13. A. G. Lecuona, On the slice-ribbon conjecture for Montesinos knots, Trans. Amer. Math. Soc. 364(1) (2012) 233–285. Crossref, Web of Science, Google Scholar
14. P. Lisca, Lens spaces, rational balls and the ribbon conjecture, Geom. Topol. 11 (2007) 429–472. Crossref, Web of Science, Google Scholar
15. P. Lisca, Sums of lens spaces bounding rational balls, Algebr. Geom. Topol. 7 (2007) 2141–2164. Crossref, Web of Science, Google Scholar
16. B. Owens and S. Strle, A characterization of the $ℤ^{n} \oplus ℤ (δ)$ lattice and definite nonunimodular intersection forms, Amer. J. Math. 134(4) (2012) 891–913. Crossref, Web of Science, Google Scholar
17. P. Ozsváth and Z. Szabó, On the Floer homology of plumbed three-manifolds, Geom. Topol. 7 (2003) 185–224 (electronic). Crossref, Web of Science, Google Scholar
18. P. Ozsváth and Z. Szabó, Absolutely graded Floer homologies and intersection forms for four-manifolds with boundary, Adv. Math. 173(2) (2003) 179–261. Crossref, Web of Science, Google Scholar
19. P. Ozsváth and Z. Szabó, On the Heegaard Floer homology of branched double-covers, Adv. Math. 194(1) (2005) 1–33. Crossref, Web of Science, Google Scholar