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Splitting Numbers of Links

  • Jae Choon Cha (a1) (a2), Stefan Friedl (a3) and Mark Powell (a4)
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.

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1. Adams, C. C., Splitting versus unlinking, J. Knot Theory Ram. 5(3) (1996), 295299.
2. Batson, J. and Seed, C., A link splitting spectral sequence in Khovanov homology, Duke Math. J. 164(5) (2015), 801841.
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. 4661 (American Mathematical Society, Providence, RI, 1978).
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. 181199 (Birkhäuser, 1986).
5. Cha, J. C. and Kim, T., Covering link calculus and iterated Bing doubles, Geom. Topol. 12(4) (2008), 21732201.
6. Cha, J. C. and Livingston, C., KnotInfo: table of knot invariants, available at http://www.indiana.edu/~knotinfo/.
7. Cha, J. C. and Livingston, C., LinkInfo: table of link invariants, available at http://www.indiana.edu/~linkinfo/.
8. Herald, C., Kirk, P. and Livingston, C., Metabelian representations, twisted Alexander polynomials, knot slicing, and mutation, Math. Z. 265(4) (2010), 925949.
9. Hillman, J. A., Algebraic invariants of links, Series on Knots and Everything, Volume 32 (World Scientific, 2002).
10. Hillman, J. A. and Sakuma, M., On the homology of finite abelian coverings of links, Can. Math. Bull. 40 (1997), 309315.
11. Kawauchi, A., A survey of knot theory (Birkhäuser, 1996).
12. Kirk, P. and Livingston, C., Twisted Alexander invariants, Reidemeister torsion, and Casson–Gordon invariants, Topology 38(3) (1999), 635661.
13. Kohn, P., Two-bridge links with unlinking number one, Proc. Am. Math. Soc. 113(4) (1991), 11351147.
14. Kohn, P., Unlinking two component links, Osaka J. Math. 30(4) (1993), 741752.
15. Murasugi, K., On a certain numerical invariant of link types, Trans. Am. Math. Soc. 117 (1967), 387422.
16. Rasmussen, J., Khovanov homology and the slice genus, Invent. Math. 182(2) (2010), 419447.
17. Shimizu, A., The complete splitting number of a lassoed link, Topol. Applic. 159(4) (2012), 959965.
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Proceedings of the Edinburgh Mathematical Society
  • ISSN: 0013-0915
  • EISSN: 1464-3839
  • URL: /core/journals/proceedings-of-the-edinburgh-mathematical-society
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