On asymmetric hyperbolic L-space knots of braid index four

TL;DR

The paper advances the study of asymmetric L-space knots by constructing an infinite family of asymmetric hyperbolic $L$-space knots with braid index four, extending the known example $t12533$. It uses a twist-family approach, starting from a base link $K_0\cup c$ (with $K_0=m239$) and obtaining $K_n$ via $(-1/n)$ surgery on $c$, showing that, under suitable $r$, all $K_n$ are $L$-space knots; cusp and length analyses along with Dehn-filling results then yield hyperbolicity and asymmetry for the family. The main contributions are an explicit infinite construction, a general criterion for twist families preserving the $L$-space property, and computational verification of hyperbolicity and asymmetry across the family, significantly expanding the set of known asymmetric $L$-space knots of braid index $4$.

Abstract

A knot is called an L-space knot if it admits a positive Dehn surgery yielding an L-space. In the SnapPy census, there are exactly 9 asymmetric L-space knots. Among them, the knot t12533 is the only known example of braid index 4. We generalize this knot, and give the first infinite family of asymmetric hyperbolic L-space knots of braid index 4.

Figures (3)

• Figure 1: The link $K_0\cup c$. A box with integer $5$ contains right handed $5$ half twists.
• Figure 2: The surgery diagram $(k\cup c\cup c_1\cup c_2)(-7,0,-\frac{1}{2},-\frac{1}{2})$ represents $(K_0\cup c)(29,0)$. A box with integer $i$ contains vertical right handed $i$ half twists.
• Figure 3: A series of handle slides and moves.

Theorems & Definitions (13)

• Theorem 1.1
• Theorem 1.2
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• Lemma 3.1
• proof