Persistent foliarity of (1,1) almost L-space knots

TL;DR

This work tackles the question of whether (1,1) non-L-space knots in $S^3$ and lens spaces are persistently foliar by constructing a branched-surface model from reduced (1,1) diagrams. The authors prove a key implication: if the branched surface fully carries a lamination, then the knot is persistently foliated, and they verify this lamination-carrying condition for (1,1) almost L-space knots. Through a detailed diagrammatic analysis and a splitting process, they show that the modified Heegaard branched surface fully carries laminations for these knots, yielding co-oriented taut foliations realizing all boundary slopes except the meridian. The results provide evidence supporting the knot L-space conjecture in this class and offer a diagrammatic route to detect persistent foliations with potential extensions to broader (1,1) non-L-space knots.

Abstract

We propose a branched surface model that could be used to prove (1,1) non-L-space knots in $S^3$ and lens spaces are persistently foliar. We verify that this model works for (1,1) almost L-space knots.

Figures (14)

• Figure 1: Reduced (1,1)-diagram parametrized by $(p,q,r,s)$
• Figure 2: Branched surface and its regular neighborhood
• Figure 3: Diamond notations
• Figure 4: Reversing the co-orientation of a source sector
• Figure 5: Modified Heegaard branched surface

Theorems & Definitions (23)

• Definition 1.1: delman2020taut, Definition 1.7
• Conjecture 1.2: delman2020taut, Conjecture 1.9
• Theorem 1.4
• Theorem 2.1: greene2018space, Theorem 1.2
• Remark 2.4
• Definition 2.5
• Remark 2.6
• Lemma 2.7
• proof
• Proposition 2.8