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Boundary of the integer-framed knot surgery trace

Statement

Assume ${A C}_{ω}$ . Let $K$ be a smooth oriented knot in the oriented $3$ -sphere with a chosen exterior $E_{K}$ and peripheral basis $(μ, λ)$ , where $λ$ is preferred. For $n \in Z$ , attach one smooth $4$ -dimensional $2$ -handle $D^{2} \times D^{2}$ to the $4$ -ball $B^{4}$ along $S^{1} \times D^{2}$ in $\partial B^{4} = S^{3}$ by the standard orientation-compatible handle attachment, using a product framing whose attaching parallel on $\partial E_{K}$ has slope $n [μ] + [λ]$ . Denote the resulting oriented trace by $X_{n} (K)$ . After rounding corners, its oriented boundary is the filling $E_{K} (n / 1)$ . This theorem covers integer framings only: a rational $p / q$ filling has a trace only after specifying an integral continued-fraction handle chain or a separately defined rational-surgery cobordism; an arbitrary rational coefficient is not the framing of a single smooth $2$ -handle.

Statement references

Smooth oriented knots and the ambient orientation

Countable-choice hypothesis for the chosen tubular data

Slope and filling convention used in this statement

Chosen knot exterior and boundary torus

Positive meridian and degree-one preferred longitude

Saveliev, Lecture 2 §2.4, Theorem 2.4 and its boundary decomposition proof

Ozsváth–Stipsicz–Szabó, Exercise 1.2.11, 2-handle attachment and boundary surgery

Facts & Assumptions

Given: The oriented $4$ -ball, a framed attaching knot $K \subset S^{3}$ , the specified product identification of its tubular neighborhood, and the slope convention for $E_{K} (n / 1)$ .

[F1]

Oriented slopes and knot Dehn filling: the filling replaces the knot neighborhood by $D^{2} \times S^{1}$ whose meridian maps to $n [μ] + [λ]$ , and its orientation agrees with the exterior orientation.

[A1]

${A C}_{ω}$ is included for the chosen exterior and peripheral-system interface The Axiom of Countable Choice ( ${A C}_{ω}$ ). The product framing is part of the handle-attachment data; after those data are supplied, the boundary decomposition makes no further choice.

Proof

1.1givenA1construct

The attaching region of the $2$ -handle is $S^{1} \times D^{2} \subset \partial (D^{2} \times D^{2})$ . It is glued to the tubular neighborhood $N (K) \subset S^{3}$ , so the portion of $\partial B^{4}$ left in the boundary after attachment is exactly $E_{K} = S^{3} ∖ int N (K)$ .

2.1step 1.1A1F1construct

The free face of the handle boundary is $D^{2} \times S^{1}$ , a solid torus. Its meridian disk $D^{2} \times {z_{0}}$ has boundary $\partial D^{2} \times {z_{0}}$ ; the specified product framing sends this curve to the attaching parallel of slope $n [μ] + [λ]$ on $\partial E_{K}$ . Thus the outgoing boundary is the union of $E_{K}$ and a solid torus with precisely the meridian image in [F1].

3.1givenA1F1step 2.1construct∎

The orientation-compatible handle attachment identifies the oriented attaching region with $N (K)$ by a boundary-orientation-reversing map. The free face $D^{2} \times S^{1}$ inherits its boundary orientation from the oriented handle, so its gluing to $E_{K}$ reverses the induced orientations on the common torus; this is exactly the oriented filling convention in [F1]. Rounding corners changes the boundary only inside a collar and preserves its orientation. Thus $\partial X_{n} (K)$ is the stated oriented filling.

Diagram

The left decomposition is the original boundary of the $4$ -ball. The handle removes the neighborhood $N (K)$ from that boundary and contributes its free face $D^{2} \times S^{1}$ ; the meridian of this new solid torus is attached along $n μ + λ$ . The diagram records the boundary pieces, while the proof above establishes the identification without using the picture.

Depends on

Used by

Worked cases of Dehn surgery on the unknot Example

Dependency tree · two levels

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Sources

Nikolai Saveliev, Lectures on the Topology of 3-Manifolds, 2nd revised edition, Lecture 2 §2.4, Theorem 2.4 (standard reference, not scraped)
Peter S. Ozsváth, András I. Stipsicz, and Zoltán Szabó, Heegaard Floer homology, version 2025-12-15, Exercise 1.2.11 (standard reference, not scraped)