Satellite Knot

A satellite knot is obtained by placing a knot inside a solid torus and embedding in so that its core is a nontrivial knot , called a companion knot. To exclude trivial cases, must not be contained in a ball inside or be isotopic within to its core (Malyutin 2016, p. 6). Satellite knots need not be prime. For example, a knot sum of two nontrivial knots is a composite knot and a satellite of either summand (Malyutin 2016, p. 2). The illustration above shows a satellite of the trefoil knot. All prime satellite knots of crossing number at most 16 are satellites of the trefoil knot, are chiral, and have wrapping number 2 (Hoste et al. 1998, pp. 42-43).

The numbers of prime satellite knots with crossing number for , 2, ... are 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 6, 10, ... (OEIS A051765). Thus the least crossing number of a prime satellite knot is 13 (Hoste et al. 1998, p. 44). A knot can be checked in the Wolfram Language to see if it is a satellite knot using KnotData[knot, "Satellite"].

No prime satellite knot is an almost alternating knot (Adams et al. 1992). Hoste et al. (1998, p. 43) describe the conjecture that if a companion knot has crossing number and the satellite wraps times longitudinally around the solid torus, then the satellite cannot be projected with fewer than crossings. On this basis, they predicted at least 27 crossings for a trefoil knot satellite with wrapping number greater than 2 and at least 17 crossings for a prime satellite of the figure eight knot.