Links have no characterising slopes

TL;DR

The paper proves that there is no analogue of characterising slopes for multi-component links: for any $n>1$-component link $L$ and any multi-slope $(r_1, \,dots, r_n)$, there exist infinitely many links $L_i$ with pairwise non-homeomorphic complements yet $L(r_1, \,dots, r_n) \\cong L_i(r_1, \,dots, r_n)$. The main technique extends rational handle slides to track framing changes and uses cancelling slides to keep the surgery outcome fixed while varying the link exterior. A satellite construction based on an unknot looping the slide band, together with hyperbolic companions of increasing simplicial volume, yields an infinite family of exteriors that are pairwise non-homeomorphic but share the same Dehn filling. This demonstrates that, unlike knots, multi-component links do not admit a finite characterisation of slopes by homeomorphism type of surgered manifolds, with the distinction captured by invariants like simplicial volume.

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.

Figures (13)

• Figure 1: The handle slide along a band $b$ with slope $3/2$.
• Figure 2: A Seifert surface $\Sigma'_1$ of $K'_1$.
• Figure 3: The neighbourhood $X$ of $C_{p_2,q_2}(K_2)$ and $K_2$.
• Figure 4: The band $b'$ from $K'_1$ to $K_2$.
• Figure 5: The link $L"$.

Theorems & Definitions (5)

• Theorem 1.1
• Definition 2.1
• Lemma 2.2
• proof
• proof : Proof of Theorem \ref{['thm:main_thm']}