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.