Hostname: page-component-586b7cd67f-2brh9 Total loading time: 0 Render date: 2024-11-26T01:22:39.009Z Has data issue: false hasContentIssue false
  • English
  • Français

Chiral smoothings of knots

Published online by Cambridge University Press:  06 November 2020

Charles Livingston
Charles Livingston*
Affiliation:
Department of Mathematics, Indiana University, Bloomington, IN47405, USA ([email protected])
Get access Rights & Permissions [Opens in a new window]

Abstract

Can smoothing a single crossing in a diagram for a knot convert it into a diagram of the knot's mirror image? Zeković found such a smoothing for the torus knot T(2, 5), and Moore–Vazquez proved that such smoothings do not exist for other torus knots T(2, m) with m odd and square free. The existence of such a smoothing implies that K # K bounds a Mobius band in B4. We use Casson–Gordon theory to provide new obstructions to the existence of such chiral smoothings. In particular, we remove the constraint that m be square free in the Moore–Vazquez theorem, with the exception of m = 9, which remains an open case. Heegaard Floer theory provides further obstructions; these do not give new information in the case of torus knots of the form T(2, m), but they do provide strong constraints for other families of torus knots. A more general question asks, for each pair of knots K and J, what is the minimum number of smoothings that are required to convert a diagram of K into one for J. The methods presented here can be applied to provide lower bounds on this number.

Keywords

KnotKnot theoryChiral smoothing
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 63 , Issue 4 , November 2020 , pp. 1048 - 1061
Check for updates
Copyright
Copyright © The Author(s), 2020. Published by Cambridge University Press on Behalf of The Edinburgh Mathematical Society

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.)

References

Abe, T. and Kanenobu, T., Unoriented band surgery on knots and links, Kobe J. Math. 31(12) (2014), 21–44.Google Scholar
Casson, A. J. and McA. Gordon, C., Cobordism of classical knots, in À la recherche de la topologie perdue, Progr. Math., , pp. 181–199, Volume 62 (Birkhäuser Boston, Boston, MA, 1986). With an appendix by P. M. Gilmer.Google Scholar
Cha, J. C. and Livingston, C., Knotinfo: table of knot invariants, http://www.indiana.edu/knotinfo (August 14, 2018)Google Scholar
Feller, P. and Krcatovich, D., On cobordisms between knots, braid index, and the upsilon-invariant, Math. Ann. 369(1–2) (2017), 301–329.CrossRefGoogle Scholar
Gilmer, P. M., On the slice genus of knots, Invent. Math. 66(2) (1982), 191–197.CrossRefGoogle Scholar
Gilmer, P. M., Slice knots in S 3, Quart. J. Math. Oxford Ser. (2) 34(135) (1983), 305–322.CrossRefGoogle Scholar
Gilmer, P. M. and Livingston, C., The nonorientable 4-genus of knots, J. Lond. Math. Soc. (2) 84(3) (2011), 559–577.CrossRefGoogle Scholar
Gordon, C. McA., Litherland, R. A. and Murasugi, K., Signatures of covering links, Can. J. Math. 33(2) (1981), 381–394.CrossRefGoogle Scholar
Hoste, J., Nakanishi, Y. and Taniyama, K., Unknotting operations involving trivial tangles, Osaka J. Math. 27(3) (1990), 555–566.Google Scholar
Ichihara, K., Jong, I. D. and Masai, H., Cosmetic banding on knots and links, ArXiv e-prints (February 2016), available at arxiv.org/abs/1602.01542.Google Scholar
Ichihara, K., Ito, T. and Saito, T., Chirally cosmetic surgeries and Casson invariants, ArXiv e-prints (July 2017), available at arxiv.org/abs/1707.00152.Google Scholar
Jabuka, S. and Van Cott, C. A., On a nonorientable analogue of the milnor conjecture, ArXiv e-prints (September 2018), available at arxiv.org/abs/1809.01779.Google Scholar
Kanenobu, T., Band surgery on knots and links, J. Knot Theory Ramifications 19(12) (2010), 1535–1547.CrossRefGoogle Scholar
Kanenobu, T., H(2)-Gordian distance of knots, J. Knot Theory Ramifications 20(6) (2011), 813–835.CrossRefGoogle Scholar
Kanenobu, T., Band surgery on knots and links, III, J. Knot Theory Ramifications 25(10) (2016), 1650056, 12.CrossRefGoogle Scholar
Kanenobu, T. and Miyazawa, Y., H(2)-unknotting number of a knot, Commun. Math. Res. 25(5) (2009), 433–460.Google Scholar
Kirk, P. and Livingston, C., Twisted knot polynomials: inversion, mutation and concordance, Topology 38(3) (1999), 663–671.CrossRefGoogle Scholar
Levine, J., Invariants of knot cobordism, Invent. Math. 8 (1969), 98–110; addendum, ibid. 8 (1969), 355.CrossRefGoogle Scholar
Lickorish, W. B. R., Unknotting by adding a twisted band, Bull. London Math. Soc. 18(6) (1986), 613–615.CrossRefGoogle Scholar
Litherland, R. A., Cobordism of satellite knots, in Four-manifold theory (Durham, N.H., 1982), pp. 327–362. Contemp. Math., Volume 35 (Amer. Math. Soc., Providence, RI, 1984)CrossRefGoogle Scholar
Livingston, C., Knots which are not concordant to their reverses, Quart. J. Math. Oxford Ser. (2) 34(35) (1983), 323–328.CrossRefGoogle Scholar
Massey, W. S., Proof of a conjecture of Whitney, Pacific J. Math. 31 (1969), 143–156.CrossRefGoogle Scholar
Moore, A. H. and Vazquez, M., A note on band surgery and the signature of a knot, (2018), available at arxiv.org/abs/1806.02440Google Scholar
Moore, A. H. and Vazquez, M., Recent advances on the non-coherent band surgery model for site-specific recombination, (2018), available at arxiv.org/abs/1810.08751Google Scholar
Murakami, H. and Yasuhara, A., Four-genus and four-dimensional clasp number of a knot, Proc. Amer. Math. Soc. 128(12) (2000), 3693–3699.CrossRefGoogle Scholar
Murasugi, K., On a certain numerical invariant of link types, Trans. Amer. Math. Soc. 117 (1965), 387–422.CrossRefGoogle Scholar
Ozsváth, P. S., Stipsicz, A. I. and Szabó, Z., Unoriented knot Floer homology and the unoriented four-ball genus, Int. Math. Res. Not. IMRN 17 (2017), 5137–5181.Google Scholar
Tristram, A. G., Some cobordism invariants for links, Proc. Cambridge Philos. Soc. 66 (1969), 251–264.CrossRefGoogle Scholar
Zeković, A., Computation of Gordian distances and H 2-Gordian distances of knots, Yugosl. J. Oper. Res. 25(1) (2015), 133–152.CrossRefGoogle Scholar