Weak Splitting Number

For a link L, the weak splitting number is the minimum number of crossing changes required to change the link into a completely split link, that is, into a collection of knots separated by embedded two-spheres. Unlike the splitting number, crossing changes can occur within a single component as well as between distinct components. The minimum is taken over all diagrams of the link.

The weak splitting number was first studied in [1]. The paper [4] completed the determination for links of 9 or fewer crossings, with the exception of L9a29, L9a30.

[1] Adams, C. Splitting versus unlinking, J. Knot Theory Ramifications, 5(3):295-299, 1996.

[2] Borodzik, M., Friedl, S., and Powell, M. Blanchfield forms and Gordian distance, J. Math. Soc. Japan, 68(3):1047-1080, 2016.

[3] Cavallo, A. and Collari, C. Slice-torus concordance invariants and Whitehead doubles of links, Canad. J. Math., page 1-31, 2018. https://arxiv.org/abs/1806.10358.

[4] Cavallo, A., Collari, C., and Conway, A. A note on the weak splitting number, 2019. http://arxiv.org/abs/1911.05677.

[5] Conway, A. Invariants of colored links and generalizations of the Burau representation, PhD thesis, Universite de Geneve, 2017.

[6] Shimizu, A. The complete splitting number of a lassoed link, Topology Appl., 159(4):959-965, 2012.