Splitting numbers and signatures
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.
Keywords
Cite
@article{arxiv.1601.07871,
title = {Splitting numbers and signatures},
author = {David Cimasoni and Anthony Conway and Kleopatra Zacharova},
journal= {arXiv preprint arXiv:1601.07871},
year = {2016}
}
Comments
13 pages, 9 figures; v2: addition of Example 2.1 and expansion of Theorem 4.7; v3: minor changes, to appear in Proc. Amer. Math. Soc; v4: a couple of typos corrected, final version