Satellite knot

A satellite knot $P(K)$ is obtained by embedding a patterned solid torus into a tubular neighborhood of a companion knot $K$.

A pattern is a knot $P$ inside a solid torus. Embedding that torus as a tubular neighborhood of $K$ produces the satellite $P(K)$.

The winding number is the integer represented by the pattern in the first homology of the solid torus, equivalently its algebraic intersection number with a meridional disk.

The $(p,q)$-cable of $K$ uses the $(p,q)$ torus knot as a pattern in a tubular neighborhood of $K$.

For a pattern of winding number $w$,whenever the relevant signatures are defined by nonsingular forms. The usual averaged convention extends the identity through roots.

If a satellite knot has pattern $P$, companion $K$, and winding number $w$, then

A Whitehead double is the satellite formed with the Whitehead pattern, which has winding number zero and whose image on the unknot is an unknot. The Satellite formula for the Alexander polynomial therefore gives $\Delta (t)\doteq 1$ for every untwisted Whitehead double.