Homomorphism obstructions for satellite maps

Abstract

A knot in a solid torus defines a map on the set of (smooth or topological) concordance classes of knots in S 3 S^3 . This set admits a group structure, but a conjecture of Hedden suggests that satellite maps never induce interesting homomorphisms: we give new evidence for this conjecture in both categories. First, we use Casson-Gordon signatures to give the first obstruction to a slice pattern inducing a homomorphism on the topological concordance group, constructing examples with every winding number besides ± 1 \pm 1 . We then provide subtle examples of satellite maps which map arbitrarily deep into the n n -solvable filtration of Cochran, Orr, and Teichner [Ann. of Math. (2) 157 (2003), pp. 433–519], act like homomorphisms on arbitrary finite sets of knots, and yet which still do not induce homomorphisms. Finally, we verify Hedden’s conjecture in the smooth category for all small crossing number satellite operators but one.

Slice knots and knot concordance

“…Modern tools from Heegaard-Floer homology can be used to obstruct satellite operators on C diff from being homomorphisms [Lev16,Hed09]. In the topological category, obstructions can be obtained from Casson-Gordon invariants [Mil23]. Most generally, we have the following conjecture.…”

Linking Numbers in Three-Manifolds

“…It is implicit in [15,14] that alternating cabling patterns do not induce homomorphisms on concordance, and also that the linking number obstruction vanishes for these patterns. Indeed, if P (U ) is isotopic to −P (U ) inside S 1 × D 2 , then the linking numbers between lifts of η to a cyclic branched cover of P (U ) must be zero: on the one hand, mirroring changes the sign of these linking numbers (while the choice of orientation on the branching set leaves the numbers unchanged); on the other hand, the pattern is isotopic to its inverse.…”