The oriented connected sum is independent of its local choices

A change of knot representative already comes with an oriented ambient isotopy by [F1]. Applying that isotopy to its ball-arc data carries the cut complement and its boundary collars to the new data, so the standard gluing produces an ambient-isotopic sum. A change of radius in one local straightening chart is the isotopy that rescales the standard ball radially while preserving its oriented diameter. To preserve the entire knot, perform this rescaling by a radial vector field in a larger straightening chart, equal to the required radial field on all swept balls and cut off before the chart boundary. The chart meets the knot only in its diameter line, to which this field is tangent. Its flow therefore preserves the whole knot setwise, including the part outside the moving ball, and identifies the two cut complements and their boundary parametrizations.

For two local oriented ball-arc charts centered at the same point, express them in one smooth coordinate chart. Their transition germ fixes the center, carries the positive tangent line of the knot to itself, and has positive determinant. Shrink to a common ball and use the smooth rescaling homotopy $G_s(x)=s^{-1}(G(sx)-G(0))$ for $s>0$, with $G_0(x)=D{G}_0(x)$. Taylor's formula gives the smooth extension at $s=0$. On a sufficiently small common ball the differentials stay uniformly close to the injective linear map $D{G}_0$, so every $G_s$ is an embedding and preserves the oriented diameter. The subgroup of invertible linear maps preserving that oriented line and the ambient orientation is path-connected: in line-normal coordinates its diagonal blocks lie in $(0,\mathrm{\infty })$ and $G{L}^{+}(2,\mathbb{R})$, and the shear block contracts linearly to zero. Thus the chart germ is isotopic through oriented ball-arc embeddings to the fixed standard chart. Here too ambient extension must preserve the whole knot. On each moving ball the isotopy velocity is tangent to the knot along its diameter. In straightening coordinates $(s,u,v)$ with knot $u=v=0$, its normal components vanish on that axis and have the form $ua+vb$, with smooth coefficients, by integrating their normal derivatives. Extend these coefficients and the tangential component slightly past the moving ball, then cut off inside the larger straightening chart. The resulting extension still has zero normal components along the knot. Finite such extensions and a partition of unity give the velocity-field construction of [F3] tangent to the whole knot, with zero field outside the chart. Uniqueness of its flow keeps the knot setwise fixed while carrying the prescribed balls and their boundary parametrizations.

If the marked points differ, move the parameter point along the knot circle by a smooth isotopy of that circle. The resulting family of knot embeddings carries the short oriented subarc along a finite chain of local straightening charts; compactness of the parameter interval gives a common sufficiently small ball radius. Apply [F3] to the whole-circle isotopy; its extension carries the entire knot to itself. For the chart isotopies use the knot-preserving extensions explicitly constructed in steps 1.1 and 1.2. The resulting ambient isotopies identify the cut complements with their boundary collars and preserve the fixed endpoint pairing in the standard gluing. Hence they induce an oriented ambient isotopy between the two connected-sum outputs.