Left-orderable surgeries of $(-2, 3, 2n+1)$-pretzel knots

TL;DR

The paper advances the left-orderability side of the L-space conjecture for a family of pretzel knots by constructing continuous families of elliptic $\mathrm{SL}_2(\mathbb{R})$-representations along unit-circle roots of the Alexander polynomial. By parameterizing $\mathrm{SL}_2(\mathbb{C})$-representations via trace coordinates and Chebyshev polynomials, it identifies a two-branch path $\rho_\pm(\theta)$ on $[\theta_n,\beta_n]$ with $|L|=1$ on the boundary and $L=1$ at an endpoint, enabling controlled boundary behavior. The key move is lifting elliptic boundary representations to $\widetilde{\mathrm{SL}}_2(\mathbb{R})$, whose left-orderability transfers to the surgered manifold’s fundamental group, proving left-orderability for slopes $\frac{m}{\ell} < 2\left\lfloor\frac{2n+4}{3}\right\rfloor$ for $n\ge 3$, $n\neq 4$. The approach clarifies the extent to which SL$_2$-representation techniques can realize the L-space conjecture in this family, while noting an unresolved range near the L-space threshold and an exceptional case at $n=4$.

Abstract

We show that the fundamental group of the $3$-manifold obtained from the $3$-sphere by $\frac{m}{l}$-surgery along the $(-2,3,2n+1)$-pretzel knot, where $n \ge 3$ is an integer and $n \not= 4$, is left-orderable if $\frac{m}{l}< 2 \lfloor \frac{2n+4}{3} \rfloor$.

Figures (1)

• Figure 1: The $(-2,3,2n+1)$-pretzel knot.

Theorems & Definitions (31)

• Theorem 1
• Lemma 2.1
• Lemma 2.2
• Lemma 2.3
• Proposition 2.4
• Proposition 3.1
• proof
• Lemma 3.2
• proof
• Lemma 3.3