Rank-expanding satellite operators on the topological knot concordance group

2022/11/08 by Livingston, Charles
#57K10 #FOS: Mathematics #Geometric Topology (math.GT)

paper · doi:10.48550/arxiv.2211.04272

Abstract

Given a fixed knot P in a solid torus and any knot K in S³, one can form the satellite of K with pattern P. This operation induces a self-map of the concordance group of knots in S³. It has been proved by Dai, Hedden, Mallick, and Stoffregen that in the smooth category there exist P for which this function is rank-expanding; that is, for some K, the set P(nK) generates an infinite rank subgroup. Here we demonstrate that similar examples exist in the case of the topological locally flat concordance group. Such examples cannot exist in the algebraic concordance group.

Related

• Iterated satellite operators on the knot concordance group
• Rank-expanding satellites, Whitehead doubles, and Heegaard Floer homology
• Homomorphism obstructions for satellite maps
• Concordance invariants from higher order covers
• A partial resolution of Hedden's conjecture on satellite homomorphisms