(1,1) almost L-space knots

TL;DR

The paper develops a diagrammatic characterization for almost L-space $(1,1)$ knots in $S^3$ and lens spaces by extending the Greene-Lewallen-Vafaee framework. It introduces the strongly almost coherent condition on standard $(1,1)$ diagrams and proves it is equivalent to the almost L-space property, with the exceptional Spin$^c$ structure of $CFK^{\infty}$ being a box plus staircase. The authors construct a new infinite family $K_j=(7+4j,3,4j,2)$ of almost L-space $(1,1)$ knots not concordant to L-space knots, and provide an algorithm to test the diagrammatic condition in practice. These results deepen the understanding of the relationship between diagrammatic data and knot Floer homology, and yield concrete tools for classifying almost L-space $(1,1)$ knots and their surgeries.

Abstract

We give a diagrammatic characterization of the $(1,1)$ knots in the three-sphere and lens spaces which admit large Dehn surgeries to manifolds with Heegaard Floer homology of next-to-minimal rank. This is inspired by a corresponding result for $(1,1)$ knots which admit large Dehn surgeries to manifolds with Heegaard Floer homology of minimal rank due to Greene-Lewallen-Vafaee.

Figures (14)

• Figure 1: The standard $(1,1)$ diagram for the four-tuple $(p,q,r,s)$ by Ras_homology, where $q,r\geq 0,$$2q + r \leq p$ and $0\leq s <p$. There are $q$ strands of "rainbow arcs" on both sides. If we label the intersection points on the top and bottom sides from left to right, then the $i$-th point on the top is identified with the $(i+s)$-th point on the bottom.
• Figure 2: Two possible cases in Definition \ref{['def: strongly_almost_coherent']}. The inconsistent arcs are the outermost arcs in the diagram on the right.
• Figure 3: The knot diagram of $K_j$ for $j\geq 0$. Here $-j$ indicates $j$ left-handed full twists.
• Figure 4: Two examples of positive almost staircases drawn in the $(i,j)$ plane. The almost staircase in (a) is of length $2$ and the staircase in (b) is of length $3$. Solid dots indicate generators and edges indicate non-trivial components of the differential (since the differential lowers the filtration level we omit the direction of the edges).
• Figure 5: The right most intersection point is a turning point.

Theorems & Definitions (44)

• Definition 1.1
• Definition 1.1
• Theorem 1.2
• Definition 1.2
• Theorem 1.3
• Corollary 1.4
• proof
• Proposition 1.4
• Definition 2.1
• Definition 2.2