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Register a new userThe crossing number of satellite knots
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Authors
Marc Lackenby
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We show that the crossing number of a satellite knot is at least 10^{-13} times the crossing number of its companion knot.
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We provide a new proof of the following results of H. Schubert: If K is a satellite knot with companion J and pattern L that lies in a solid torus T in which it has index k, then the bridge numbers satisfy the following: 1) The bridge number of K is greater than or equal to the product of k and the bridge number of J; 2) If K is a composite knot (this is the case k = 1), then the bridge number of K is one less than the sum of the bridge numbers of J and L.
We consider compact 3-manifolds M having a submersion h to R in which each generic point inverse is a planar surface. The standard height function on a submanifold of the 3-sphere is a motivating example. To (M, h) we associate a connectivity graph G. For M in the 3-sphere, G is a tree if and only if there is a Fox reimbedding of M which carries horizontal circles to a complete collection of complementary meridian circles. On the other hand, if the connectivity graph of the complement of M is a tree, then there is a level-preserving reimbedding of M so that its complement is a connected sum of handlebodies. Corollary: The width of a satellite knot is no less than the width of its pattern knot. In particular, the width of K_1 # K_2 is no less than the maximum of the widths of K_1 and K_2.
Every link in the 3-sphere has a projection to the plane where the only singularities are pairwise transverse triple points. The associated diagram, with height information at each triple point, is a triple-crossing diagram of the link. We give a set of diagrammatic moves on triple-crossing diagrams analogous to the Reidemeister moves on ordinary diagrams. The existence of n-crossing diagrams for every n>1 allows the definition of the n-crossing number. We prove that for any nontrivial, nonsplit link, other than the Hopf link, its triple-crossing number is strictly greater than its quintuple-crossing number.
We prove that the crossing changes, Delta moves, and sharp moves are unknotting operations on welded knots.
We show there exists a topologically slice knot $K$ such that the knots ${M^n(K)}_{n=0}^infty$ obtained by iterated satellite operations by the Mazur pattern span an infinite-rank summand of the smooth knot concordance group. This answers a question raised by Feller-Park-Ray.
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