A note on the weak splitting number

Abstract

The weak splitting number wsp(L) of a link L is the minimal number of crossing changes needed to turn L into a split union of knots. We describe conditions under which certain R-valued link invariants give lower bounds on wsp(L). This result is used both to obtain new bounds on wsp(L) in terms of the multivariable signature and to recover known lower bounds in terms of the τ and s-invariants. We also establish new obstructions using link Floer homology and apply all these methods to compute wsp for all but two of the 130 prime links with 9 or fewer crossings.2010 Mathematics Subject Classification. 57M27 (57M25). This work started when the second-named author was visiting the first-named author in Bonn. All three authors are grateful to the Max Plank Institute in Bonn for its support and hospitality. We thank Paolo Lisca and Chuck Livingston for helpful discussions. We are indebted to an anonymous referee for pointing out a mistake in a previous version of this paper, and for their valuable comments.

Slice‐torus link invariants, combinatorial invariants and positivity conditions

“…Now, a quick inspection of shows that can be unlinked with crossing changes on a single component (cf. [11, Proposition 2.7]), therefore . Since , by [10, Proposition 2.11], we obtain the desired equality.…”

“…Proof. The equality in (i) has been proved in [11,Remark 2.3]. Consider a minimal unlinking sequence for L with m mixed crossing changes and s self-crossing changes.…”

Slice-torus link invariants, combinatorial invariants, and positivity conditions

“…Proof. The equality in (i) has been proved in [9,Remark 2.3]. Consider a minimal unlinking sequence for L with m mixed crossing changes and s self crossing changes.…”