We study when an essential tangle decomposition of a satellite knot gives an essential tangle decomposition of the companion knot, that is, when the decomposing sphere can be isotoped to intersect the knotted solid torus identified with the pattern in meridian disks.
@article{urn:eudml:doc:44402, title = {Tangle decompositions of satellite knots.}, journal = {Revista Matem\'atica de la Universidad Complutense de Madrid}, volume = {12}, year = {1999}, pages = {417-437}, zbl = {0958.57008}, mrnumber = {MR1740467}, language = {en}, url = {http://dml.mathdoc.fr/item/urn:eudml:doc:44402} }
Hayashi, Chuichiro; Matsuda, Hiroshi; Ozawa, Makoto. Tangle decompositions of satellite knots.. Revista Matemática de la Universidad Complutense de Madrid, Tome 12 (1999) pp. 417-437. http://gdmltest.u-ga.fr/item/urn:eudml:doc:44402/