Satellite Knot

Let  be a knot inside a torus, and knot the torus in the shape of a second knot (called the companion knot) , with certain additional mild restrictions to avoid trivial cases. Then the new knot resulting from  is called the satellite knot . All satellite knots are prime (Hoste et al. 1998). The illustration above illustrates a satellite knot of the trefoil knot, which is the form all satellite knots of 16 or fewer crossings take (Hoste et al. 1998). Satellites of the trefoil share the trefoil's chirality, and all have wrapping number 2.

Any satellite knot having wrapping number  must have at least 27 crossings, and any satellite of the figure eight knot must have at least 17 crossings (Hoste et al. 1998). The numbers of satellite knots with  crossings are 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 6, 10, ... (OEIS A051765), so the satellite knot of minimal crossing number occurs for 13 crossings. A knot can be checked in the Wolfram Language to see if it is a satellite knot using KnotData[knot, "Satellite"] (although all knots currently implemented in the Wolfram Language are nonsatellite knots).

No satellite knot is an almost alternating knot. If a companion knot has crossing number  and the satellite ravels  times longitudinally around the solid torus, then it is conjectured that the satellite cannot be projected with fewer than  crossings (Hoste et al. 1998).

REFERENCES:

Adams, C. C. The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots. New York: W. H. Freeman, pp. 115-118, 1994.

Hoste, J.; Thistlethwaite, M.; and Weeks, J. "The First  Knots." Math. Intell. 20, 33-48, Fall 1998.

Sloane, N. J. A. Sequence A051765 in "The On-Line Encyclopedia of Integer Sequences."