Satellite knot

Definition

Let $V = S^1 \times D^2$ be the solid torus, and let $P \subset \operatorname{int} V$ be a tame knot, the pattern, such that:

$P$ is not contained in any $3$ -ball in $V$ ;
$P$ is not isotopic in $V$ to the core circle $S^1 \times \set{0}$ .

Let $C \subset S^3$ be a knot that is not the unknot, the companion. Let $h \colon V \to S^3$ be an embedding onto a closed tubular neighbourhood of $C$ . A knot in $S^3$ is a satellite knot if it is isotopic to $h(P)$ for some such $P$ , $C$ and $h$ .

Remarks

Equivalently, a knot $K$ is a satellite knot if $S^3 \setminus K$ contains an incompressible torus that is not parallel to the boundary of a neighbourhood of $K$ . Example: let $C$ be a trefoil knot and let $P$ be the $(2, 1)$ -curve on a smaller concentric torus in $V$ . Then $h(P)$ is a cable of the trefoil and a satellite knot. Non-example: if $P$ is the core circle, clause 2 fails and $h(P)$ is just $C$ . Every knot sum $K_1 \# K_2$ of nontrivial knots is a satellite with companion $K_1$ . Whitehead doubles are satellites whose pattern has winding number $0$ in $V$ .

Reading path

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Knot concept
Unknot example
Satellite knot definition

Tag	`5KGKGTW2XY`
URI	`m2://5KGKGTW2XY`
Kind	definition
Writing	W0 stub
MSC	57K10
Defines	Satellite knot
Uses	Knot, Unknot
Changed	2 November 2026

Satellite knot

Definition

Remarks

Reading path

Further reading

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