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Splitting Numbers of Links

Part of: Topological manifolds Low-dimensional topology

Published online by Cambridge University Press:  03 January 2017

Jae Choon Cha ,
Stefan Friedl  and
Mark Powell
Jae Choon Cha
Affiliation:
Department of Mathematics, POSTECH, Pohang 790–784, Republic of Korea (jccha@postech.ac.kr) School of Mathematics, Korea Institute for Advanced Study, Seoul 130–722, Republic of Korea
Stefan Friedl
Affiliation:
Mathematisches Institut, Universität zu Köln, 50931 Köln, Germany (sfriedl@gmail.com)
Mark Powell
Affiliation:
Département de Mathématiques, Université du Québec à Montréal, Montréal, QC, Canada (mark@cirget.ca)
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Abstract

The splitting number of a link is the minimal number of crossing changes between different components required, on any diagram, to convert it to a split link. We introduce new techniques to compute the splitting number, involving covering links and Alexander invariants. As an application, we completely determine the splitting numbers of links with nine or fewer crossings. Also, with these techniques, we either reprove or improve upon the lower bounds for splitting numbers of links computed by Batson and Seed using Khovanov homology.

Keywords

splitting numbers of linkscovering linksAlexander polynomial

MSC classification

Secondary: 57M25: Knots and links in $S^3$ 57M27: Invariants of knots and 3-manifolds 57N70: Cobordism and concordance
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 60 , Issue 3 , August 2017 , pp. 587 - 614
Copyright
Copyright © Edinburgh Mathematical Society 2017 

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References

1. Adams, C. C., Splitting versus unlinking, J. Knot Theory Ram. 5(3) (1996), 295–299.Google Scholar
2. Batson, J. and Seed, C., A link splitting spectral sequence in Khovanov homology, Duke Math. J. 164(5) (2015), 801–841.Google Scholar
3. Casson, A. and Gordon, C. McA., On slice knots in dimension three, in Algebraic and geometric topology, Part 2, Proceedings of Symposia in Pure Mathematics, Volume 32, pp. 46–61 (American Mathematical Society, Providence, RI, 1978).Google Scholar
4. Casson, A. and Gordon, C. McA., Cobordism of classical knots, in A la recherche de la topologie perdue, Progress in Mathematics, Volume 62, pp. 181–199 (Birkhäuser, 1986).Google Scholar
5. Cha, J. C. and Kim, T., Covering link calculus and iterated Bing doubles, Geom. Topol. 12(4) (2008), 2173–2201.Google Scholar
6. Cha, J. C. and Livingston, C., KnotInfo: table of knot invariants, available at http://www.indiana.edu/~knotinfo/.Google Scholar
7. Cha, J. C. and Livingston, C., LinkInfo: table of link invariants, available at http://www.indiana.edu/~linkinfo/.Google Scholar
8. Herald, C., Kirk, P. and Livingston, C., Metabelian representations, twisted Alexander polynomials, knot slicing, and mutation, Math. Z. 265(4) (2010), 925–949.CrossRefGoogle Scholar
9. Hillman, J. A., Algebraic invariants of links, Series on Knots and Everything, Volume 32 (World Scientific, 2002).Google Scholar
10. Hillman, J. A. and Sakuma, M., On the homology of finite abelian coverings of links, Can. Math. Bull. 40 (1997), 309–315.Google Scholar
11. Kawauchi, A., A survey of knot theory (Birkhäuser, 1996).Google Scholar
12. Kirk, P. and Livingston, C., Twisted Alexander invariants, Reidemeister torsion, and Casson–Gordon invariants, Topology 38(3) (1999), 635–661.Google Scholar
13. Kohn, P., Two-bridge links with unlinking number one, Proc. Am. Math. Soc. 113(4) (1991), 1135–1147.Google Scholar
14. Kohn, P., Unlinking two component links, Osaka J. Math. 30(4) (1993), 741–752.Google Scholar
15. Murasugi, K., On a certain numerical invariant of link types, Trans. Am. Math. Soc. 117 (1967), 387–422.Google Scholar
16. Rasmussen, J., Khovanov homology and the slice genus, Invent. Math. 182(2) (2010), 419–447.Google Scholar
17. Shimizu, A., The complete splitting number of a lassoed link, Topol. Applic. 159(4) (2012), 959–965.Google Scholar