$L$-space knots with positive surgeries that are not weakly symplectically fillable

TL;DR

This paper develops a method to identify $L$-space knots with positive surgeries that cannot be weakly symplectically filled, providing new examples of hyperbolic $L$-spaces lacking such fillability.

Contribution

It introduces a general strategy to detect non-fillability of $L$-spaces using arithmetic invariants and presents an infinite family of hyperbolic $L$-spaces without weak symplectic fillings.

Findings

Computed geometric invariants obstructing fillability.

Identified an infinite family of hyperbolic $L$-spaces without weak fillings.

Demonstrated the existence of $L$-spaces outside the weakly fillable class.

Abstract

In this paper we discuss a general strategy to detect the absence of weakly symplectic fillings of $L$-spaces. We start from a generic $L$-space knot and consider (positive) Dehn surgeries on it. We compute, using arithmetic data depending only on the knot type and the surgery coefficient, the value of the relevant geometric invariants used to obstruct fillability. We also provide a new example of an infinite family of hyperbolic $L$-spaces that do not admit weakly symplectic fillings. These are manifolds that lie inside $\{\text{Tight}\}$ but not inside $\{\text{Weakly Fillable}\}$.