Splitting numbers and signatures

Cimasoni, David; Conway, Anthony; Zacharova, Kleopatra

doi:10.1090/proc/13156

Splitting numbers and signatures
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by David Cimasoni, Anthony Conway and Kleopatra Zacharova PDF

Proc. Amer. Math. Soc. 144 (2016), 5443-5455 Request permission

Abstract:

The splitting number of a link is the minimal number of crossing changes between different components required to convert it into a split link. We obtain a lower bound on the splitting number in terms of the (multivariable) signature and nullity. Although very elementary and easy to compute, this bound turns out to be suprisingly efficient. In particular, it makes it a routine check to recover the splitting number of 129 out of the 130 prime links with at most 9 crossings. Also, we easily determine 16 of the 17 splitting numbers that were studied by Batson and Seed using Khovanov homology, and later computed by Cha, Friedl and Powell using a variety of techniques. Finally, we determine the splitting number of a large class of 2-bridge links which includes examples recently computed by Borodzik and Gorsky using a Heegaard Floer theoretical criterion.

References

Colin C. Adams, Splitting versus unlinking, J. Knot Theory Ramifications 5 (1996), no. 3, 295–299. MR 1405713, DOI 10.1142/S0218216596000205
Joshua Batson and Cotton Seed, A link-splitting spectral sequence in Khovanov homology, Duke Math. J. 164 (2015), no. 5, 801–841. MR 3332892, DOI 10.1215/00127094-2881374
Maciej Borodzik, Stefan Friedl, and Mark Powell, Blanchfield forms and Gordian distance, Journal of the Mathematical Society of Japan, to appear.
Maciej Borodzik and Eugene Gorsky, Immersed concordances of links and Heegaard Floer homology, January 2016. http://arxiv.org/abs/1601.07507.
A. J. Casson and C. McA. Gordon, On slice knots in dimension three, Algebraic and geometric topology (Proc. Sympos. Pure Math., Stanford Univ., Stanford, Calif., 1976) Proc. Sympos. Pure Math., XXXII, Amer. Math. Soc., Providence, R.I., 1978, pp. 39–53. MR 520521
A. J. Casson and C. McA. Gordon, Cobordism of classical knots, À la recherche de la topologie perdue, Progr. Math., vol. 62, Birkhäuser Boston, Boston, MA, 1986, pp. 181–199. With an appendix by P. M. Gilmer. MR 900252
Jae Choon Cha, Stefan Friedl, and Mark Powell, Splitting numbers of links, Proceedings of the Edinburgh Mathematical Society, to appear.
Jae Choon Cha and Charles Livingston, Linkinfo: Table of knot invariants, January 24, 2016. http://www.indiana.edu/$\sim$linkinfo.
David Cimasoni, A geometric construction of the Conway potential function, Comment. Math. Helv. 79 (2004), no. 1, 124–146. MR 2031702, DOI 10.1007/s00014-003-0777-6
David Cimasoni and Anthony Conway, Colored tangles and signatures, July 2015. http://arxiv.org/abs/1507.07818.
David Cimasoni and Vincent Florens, Generalized Seifert surfaces and signatures of colored links, Trans. Amer. Math. Soc. 360 (2008), no. 3, 1223–1264. MR 2357695, DOI 10.1090/S0002-9947-07-04176-1
D. Cooper, The universal abelian cover of a link, Low-dimensional topology (Bangor, 1979) London Math. Soc. Lecture Note Ser., vol. 48, Cambridge Univ. Press, Cambridge-New York, 1982, pp. 51–66. MR 662427
Marc Culler, Nathan M. Dunfield, and Jeffrey R. Weeks, SnapPy, a computer program for studying the topology of $3$-manifolds, Available at http://snappy.computop.org.
Alex Degtyarev, Vincent Florens, and Ana Lecuona, The signature of a splice, September 2014. http://arxiv.org/abs/1409.5873.
Chris Herald, Paul Kirk, and Charles Livingston, Metabelian representations, twisted Alexander polynomials, knot slicing, and mutation, Math. Z. 265 (2010), no. 4, 925–949. MR 2652542, DOI 10.1007/s00209-009-0548-1
Akio Kawauchi, The Alexander polynomials of immersed concordant links, Bol. Soc. Mat. Mex. (3) 20 (2014), no. 2, 559–578. MR 3264631, DOI 10.1007/s40590-014-0023-9
Peter Kohn, Unlinking two component links, Osaka J. Math. 30 (1993), no. 4, 741–752. MR 1250780
Hisako Kondo, Knots of unknotting number $1$ and their Alexander polynomials, Osaka Math. J. 16 (1979), no. 2, 551–559. MR 539606
Marc Lackenby, Elementary knot theory, to be published by the Clay Mathematics Institute, 2014.
J. Levine, Knot cobordism groups in codimension two, Comment. Math. Helv. 44 (1969), 229–244. MR 246314, DOI 10.1007/BF02564525
Kunio Murasugi, On a certain numerical invariant of link types, Trans. Amer. Math. Soc. 117 (1965), 387–422. MR 171275, DOI 10.1090/S0002-9947-1965-0171275-5
Jacob Rasmussen, Khovanov homology and the slice genus, Invent. Math. 182 (2010), no. 2, 419–447. MR 2729272, DOI 10.1007/s00222-010-0275-6
Ayaka Shimizu, The complete splitting number of a lassoed link, Topology Appl. 159 (2012), no. 4, 959–965. MR 2876702, DOI 10.1016/j.topol.2011.11.028
A. G. Tristram, Some cobordism invariants for links, Proc. Cambridge Philos. Soc. 66 (1969), 251–264. MR 248854, DOI 10.1017/s0305004100044947