LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedverified 2026-10-22 (gpt-6.1-sol)

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A decomposing sphere realizes a connected sum of knots

Statement

Assume the Axiom of Countable Choice. Let $K$ be a smooth oriented knot in $S^{3}$ , and let $S \subset S^{3}$ be a smoothly embedded sphere transverse to $K$ in exactly two points. Cutting along $S$ gives two properly embedded arcs; cap each arc by an orientation-compatible simple arc on $S$ , pushed slightly into the corresponding boundary collar to obtain oriented knots $K_{1}$ and $K_{2}$ . Then the oriented isotopy class of $K$ is the connected sum $K_{1} # K_{2}$ .

Statement references

The connected-sum construction and orientation convention

Facts & Assumptions

Given: AC $_{ω}$ , a smooth oriented knot $K$ , and a smooth sphere $S$ transverse to $K$ in two points.

[F1]

Smooth Schoenflies supplies an ambient isotopy taking $S$ to the equator; its endpoint identifies both ball sides simultaneously, with the same equatorial boundary parametrization. (Smooth Schoenflies for an embedded two-sphere in the three-sphere)

[F2]

The fixed standard ball-arc connected-sum construction is independent of its local choices under AC $_{ω}$ . (Oriented connected sum and prime knot, The oriented connected sum is independent of its local choices, The Axiom of Countable Choice ( ${A C}_{ω}$ ))

Proof

Given: The stated knot, sphere, and AC $_{ω}$ .

1.1givenF1construct

Apply [F1] and carry the knot through the same ambient isotopy. We now have the two standard hemispherical balls, each containing one proper arc of the knot. Cap each arc by a simple boundary arc, pushed slightly into a boundary collar; orient the cap to close the oriented original arc. Boundary caps give the same capped knot class: two simple arcs in a two-sphere joining the same two points are isotopic keeping the endpoints, with no other marked points to avoid. One explicit reduction puts the arcs transverse, removes an outermost bigon successively, and then slides the resulting disjoint arcs across the disk they cobound; all moves take place in surface disks and extend to their ball collars.

2.1F1F2step 1.1construct∎

A small collar neighbourhood of each cap arc is a ball meeting the capped knot in one trivial arc. Delete these two ball-arc neighbourhoods. The remaining arcs are precisely the original two knot pieces, with endpoint collars on the common equator. Gluing by the fixed orientation-matching two-ball convention recovers $K$ ; the simultaneous boundary parametrization in [F1] makes the two sides agree. Straighten the cap collar and its positive local frame before gluing, which is a change of the local data allowed by [F2]. Hence this is the connected sum $K_{1} # K_{2}$ in the stated convention. Transporting back through the original ambient isotopy proves the assertion. No extension of an arbitrary prescribed sphere diffeomorphism is used.

Sources

The oriented connected sum is independent of its local choices

Smooth Schoenflies for an embedded two-sphere in the three-sphere

Depends on

Used by

Dependency tree · two levels

17 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Burde and Zieschang, Knots, 2nd revised and extended edition (standard reference, not scraped)
Allen Hatcher, Notes on Basic 3-Manifold Topology (standard reference, not scraped)