Links have no characterising slopes

TL;DR

This paper proves that, unlike knots, multi-component links in 3-spheres do not have characterising slopes, as infinitely many non-homeomorphic links can produce the same Dehn filling results.

Contribution

It establishes that no analogue of characterising slopes exists for multi-component links, extending the understanding of link Dehn surgeries.

Findings

No characterising slopes for multi-component links.

Existence of infinitely many links with identical Dehn fillings.

Contrasts with the knot case where characterising slopes exist.

Abstract

We show that there is no analogue of characterising slopes for multi-component links. Concretely, we show that for any ordered link L in S3 with n>1 components and any rational slopes r_1, ..., r_n, there are infinitely many links L_i with non-homeomorphic complements such that the Dehn fillings L(r_1, ..., r_n) and L_i(r_1, ..., r_n) are homeomorphic.