The complete splitting number of a lassoed link

TL;DR

This paper introduces the concept of lassoing on links, establishes bounds on the complete splitting number after multiple lassoings, and constructs examples illustrating the relationship between unlinking number and splitting number.

Contribution

It defines lassoing on links, derives bounds for the complete splitting number after r-iterated lassoings, and constructs links with specific splitting and unlinking properties.

Findings

Complete splitting number bounds for r-iterated lassoings

Construction of algebraically completely splittable links from knots

Examples where unlinking number exceeds splitting number

Abstract

In this paper, we define a lassoing on a link, a local addition of a trivial knot to a link. Let K be an s-component link with the Conway polynomial non-zero. Let L be a link which is obtained from K by r-iterated lassoings. The complete splitting number split(L) is greater than or equal to r+s-1, and less than or equal to r+split(K). In particular, we obtain from a knot by r-iterated component-lassoings an algebraically completely splittable link L with split(L)=r. Moreover, we construct a link L whose unlinking number is greater than split(L).