Characterising slopes for hyperbolic knots and Whitehead doubles

Abstract

A slope $p/q \in \mathbb{Q}$ is characterising for a knot $K \subset \mathbb{S}^3$ if the oriented homeomorphism type of the manifold $\mathbb{S}^3_K(p/q)$ obtained by Dehn surgery of slope $p/q$ on $K$ uniquely determines the knot $K$. We combine analysis of JSJ decompositions with techniques involving lengths of shortest geodesics to find explicit conditions for a slope to be characterising for $K$ in the case where $K$ is any hyperbolic knot or any satellite knot by a hyperbolic pattern. Assuming that the list of 2-cusped orientable hyperbolic 3-manifolds obtained using the computer programme SnapPy is complete up to a certain point, we use hyperbolic volume inequalities to generate a refinement for the special case of Whitehead doubles. We also construct pairs of multiclasped Whitehead doubles of double twist knots for which $1/q$ is a non-characterising slope.

Figures (8)

• Figure 1: The $n$-clasped $t$-twisted Whitehead link, $W^n_t = V^n_t \cup U$.
• Figure 2: A pair of knots, $K=W^n(T^m_q)$ and $K'=W^m(T^n_q)$, sharing a $1/q$-surgery.
• Figure 3: The twist knots for which all non-integral slopes are characterising.
• Figure 4: Rolfsen $-t$-twist giving $\mathbb{S}^3_{W^\pm}(1/t) \cong \mathbb{S}^3_{T^\pm_t}$.
• Figure 5: Rolfsen $-n$-twist giving $\mathbb{S}^3_B(1/n) \cong \mathbb{S}^3_{W^n}$.

Theorems & Definitions (69)

• Definition 1.1
• Theorem 1.2
• Corollary 1.3
• Theorem 1.4
• Corollary 1.5
• Theorem 1.6
• Theorem 1.7
• Definition 2.1
• Definition 2.2
• Definition 2.3