Non-characterizing slopes for more 3-manifolds

TL;DR

This work generalizes the notion of characterizing slopes to compact orientable 3-manifolds with torus boundary by producing infinite pairs of knots representing any given homotopy class $h$ whose orientation-preserving $0$-surgeries are homeomorphic. The method combines Myers’ hyperbolic knot realization with a generalized satellite construction using pattern knots $A_n$ and $B_n$ in a solid torus, yielding exteriors that split into two JSJ pieces yet share identical $0$-surgery outcomes. The key technical advance is showing the exteriors $M_K\cup X_n$ and $M_K\cup Y_n$ are non-homeomorphic (via distinct volumes) while the $(0,0)$-surgery on the pattern yields the same manifold, and that the satellites remain homotopic to the original knot $K$. This provides infinitely many non-characterizing examples and extends the framework of characterizing slopes through a JSJ/dehn-surgery perspective with hyperbolic geometry.

Abstract

For any homotopy class h in any compact orientable 3-manifold M which is closed or has exclusively torus boundary components, we produce infinitely many pairs of distinct knots representing h with orientation-preserving homeomorphic 0-surgeries.

Figures (3)

• Figure 1: The link $\overline{L}$ in $S^3$
• Figure 2: Surgery Diagram for $Z_n$
• Figure 3: Diagram of $A_n$

Theorems & Definitions (15)

• Theorem 1.1
• Definition 2.1
• Definition 2.2
• Definition 3.1
• Definition 3.2
• Definition 3.3
• Lemma 3.4
• proof
• Theorem 3.5
• Lemma 3.6