Oriented connected sum and prime knot

Fix a coordinate orientation-reversing reflection $r:S^3\to S^3$ for the mirror convention. Let $K$ and $J$ be smooth oriented knots in the standard oriented three-sphere. A connected-sum datum consists of one smoothly embedded closed three-ball around a short subarc of $K$ and one around a short subarc of $J$, each meeting its knot in a single trivial properly embedded arc, together with the fixed standard orientation-reversing boundary identification of the two standard oriented ball boundaries that joins the oriented arcs into one circle. Choose this standard map once in the standard two-ball model so that it commutes with $r$ and an orientation-preserving half-turn interchanging the two balls preserves the joining convention. Here a trivial arc is an embedded interval in a ball that is carried by an orientation-preserving diffeomorphism of ball-arc pairs to a straight diameter. Remove the interiors of the two balls and glue the remaining boundary spheres using this fixed identification; the resulting oriented knot in the standard oriented three-sphere is a connected sum of $K$ and $J$, denoted $K\mathrm{\#}J$ for the specified datum.

The standard unknot is the counterclockwise circle $U=\{(x_1,x_2,0)\in {\mathbb{R}}^3:x_1^2+x_2^2=1\}\subset S^3={\mathbb{R}}^3\cup \{\mathrm{\infty }\}$. Under inverse stereographic projection $x\mapsto (2x\mathrm{/}(1+\parallel x{\parallel }^2),(\parallel x{\parallel }^2-1)\mathrm{/}(1+\parallel x{\parallel }^2))$, this is the unit circle in the first coordinate two-plane of the unit sphere in ${\mathbb{R}}^4$. A knot is nontrivial if it is not ambient isotopic through orientation-preserving diffeomorphisms of $S^3$ to the standard unknot. A nontrivial knot is prime if it has no connected-sum presentation whose two summand knots are both nontrivial.

Smooth oriented knots and ambient isotopy

Independence of the local connected-sum choices

Burde and Zieschang, Chapter 7

Friedl, Proposition 2.14

The circles stand for arbitrary oriented knot diagrams outside the small balls. The short gaps are the trivial arcs removed from the two knots; the two joining arcs indicate the fixed orientation-matching gluing. This is only a construction schematic and contains no crossing data.