Satellite Knot


Let K_1 be a knot inside a torus, and knot the torus in the shape of a second knot (called the companion knot) K_2, with certain additional mild restrictions to avoid trivial cases. Then the new knot resulting from K_1 is called the satellite knot K_3. 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 >2 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 n 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 k and the satellite ravels m times longitudinally around the solid torus, then it is conjectured that the satellite cannot be projected with fewer than km^2 crossings (Hoste et al. 1998).

See also

Almost Alternating Knot, Cable Knot, Companion Knot, Composite Knot, Doubled Knot, Hyperbolic Knot, Torus Knot, Twist Knot

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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 1701936 Knots." Math. Intell. 20, 33-48, Fall 1998.Sloane, N. J. A. Sequence A051765 in "The On-Line Encyclopedia of Integer Sequences."

Referenced on Wolfram|Alpha

Satellite Knot

Cite this as:

Weisstein, Eric W. "Satellite Knot." From MathWorld--A Wolfram Web Resource.

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