Links all of whose cyclic branched covers are L-spaces

Abstract

We show that for the pretzel knots $K_k=P(3,-3,-2k-1)$, the $n$-fold cyclic branched covers are L-spaces for all $n\geq 1$. In addition, we show that the knots $K_k$ with $k\geq 1$ are quasipositive and slice, answering a question of Boileau-Boyer-Gordon. We also extend results of Teragaito giving examples of two-bridge knots with all L-space cyclic branched covers to a family of two-bridge links.

Figures (16)

• Figure 1: On the left we give a diagram for the figure-eight knot $K$ which after a planar isotopy has a rotational symmetry of order 2. Once we quotient by this symmetry we obtain the symmetric link $Q\cup A$. On the right, $L_n$ is an $n$-periodic link with quotient $A\cup Q$.
• Figure 2: The link $L(k_1,k_2,\ldots, k_n)$, where $k_1,\ldots,k_n$ are any integers. A twist box labelled $k_i$ denotes $k_i$ signed half-twists. See Figure \ref{['pretzelandsymmetry']} for an example of our signed twist conventions.
• Figure 3: The pretzel link $P(p_1,p_2,\ldots p_n)$ and a symmetry of $P(-3,3,-3)$.
• Figure 4: Let $K_k=P(-3,-2k-1,3)\cong P(3,-3, -2k-1)$. There is an isotopy of $\overline{K_k}\cup \overline{a}$ to a diagram where $\overline{a}$ sits inside a solid torus with $\overline{K_k}$ its meridian
• Figure 5: For 2-periodic link $L$, the $n$-fold branched cover can be expressed in two ways. The left-hand side of the diagram describes $\Sigma_{n,2}(\overline{L}\cup \overline{a})$ as first branching over $\overline{a}$, and then over $L$ (the lift of $\overline{L}$). On the right-hand side, we branch first over $\overline{L}$ and then over the lift of $\overline{a}$ which we denote $L_n$.

Theorems & Definitions (27)

• Theorem 1.1
• Proposition 1.3
• Theorem 1.4
• Theorem 1.5
• Proposition 2.1
• proof
• Definition 3.1
• Theorem 3.2
• Definition 4.1
• Lemma 4.2