Surgeries between lens spaces of type $L(n,1)$ and the Heegaard Floer $d$-invariant

TL;DR

The paper develops a robust $d$-invariant surgery framework for $L$-space knots and leverages the Ozsváth–Szabó mapping cone to study Dehn surgeries between lens spaces, focusing on distance one cases between $L(n,1)$ and $L(s,1)$. By combining $d$-invariants, Rasmussen's notation, and Casson–Walker obstructions, the authors classify all potential pairs $(n,s)$ compatible with distance one surgeries, translating many instances to band surgeries on torus links via double branched covers. The main contributions include a comprehensive list of admissible pairs, several explicit obstruction-based constraints, and a deeper link between geometric surgery theory and applications to DNA topology through banding of torus links. The results yield precise obstruction-driven criteria that significantly restrict possible surgeries, with a small set of exceptional cases remaining unresolved and tied to band-surgery realizations. Overall, the work advances understanding of lens-space surgeries and their connections to knot Floer theory and biological models.

Abstract

We establish a $d$-invariant surgery formula for $L$-space knots that provides an effective tool for studying surgeries between lens spaces. Using this formula, we classify distance one surgeries between lens spaces of the form $L(n,1)$. This classification has direct applications to band surgeries between torus links $T(2,n)$, with connections to DNA topology. In particular, we show that chirally cosmetic banding of torus links can possibly occur only when $n=1,5,9$ or $10$.

Figures (9)

• Figure 1: Examples of band surgeries, which lift to distance one surgeries in double branched covers.
• Figure 2: An example of a simple knot $K(5,1,2)$ in $L(5,1)$. To draw $K(5,1,2)$ in the Heegaard diagram, we place two points $x'_0$ and $x'_2$ next to $x_0$ and $x_2$ respectively, and connect them in $U_{\alpha}$ and $U_{\beta}$.
• Figure 3: The mapping cone $\mathbb{X}^+_{\mathfrak{s}}$ with $\mathfrak{s}=G_{Y_{\gamma}(K), K_{\gamma}}(\xi)$.
• Figure 4: A truncated mapping cone in Rasmussen's notation. While the full complex extends infinitely in both directions, we show only the essential portion that determines the homology.
• Figure 5: A mapping cone satisfying conditions in Proposition \ref{['Prop:L-spacesurgerycriteria']}, shown in truncated form. The complex extends infinitely with $-$ symbols to the left and $+$ symbols to the right.

Theorems & Definitions (63)

• Theorem 1.1
• Remark 1.2
• Corollary 1.3
• Corollary 1.4
• Corollary 1.5
• Theorem 1.6
• Definition 2.1
• Lemma 2.2
• Theorem 3.1
• Lemma 3.2