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 . 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. We then provide subtle examples of satellite maps which map arbitrarily deep into the n-solvable filtration of [COT03], 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 but one small crossing number satellite operator.

A Cable Knot and BPS-Series

“…A well-known example of satellite knots is a cable knot h K (P ) = C (r,s) (K ) that is obtained by choosing P to be the (r, s)-torus knot pushed into the interior of the S 1 × D 2 . This map h K has been investigated in [23,25,26].…”

Knot Floer homology of satellite knots with (1,1)-patterns

“…P (U ) = U ) that admit genus-one doublypointed Heegaard diagram. Note such patterns cannot be dealt with using the obrstruction coming from the Casson-Gordon invariant due to Miller [11].…”

“…The assumption that a pattern P induces a homomorphism on the concordance group constrains the behavior of the τ -invariant under the action by P , i.e. τ (P (K)) = |w(P )|τ (K) for any knot K (see the proof of Corollary 1.9 in Section 6 or Proposition 5.4 of [11]). This together with Theorem 1.7 implies Corollary 1.9.…”

Linking number obstructions to satellite homomorphisms

“…Many patterns can be obstructed from inducing a homomorphism by the unexciting requirement that P (U ) must be a slice knot. However, if P (U ) is slice then classical invariants such as the Alexander polynomial and the Tristram-Levine signature function cannot obstruct P from inducing a homomorphism on C [30,34]. The first interesting nonhomomorphism in the literature was therefore not identified until work of Gompf [15], who showed that the Whitehead pattern does not induce a homomorphism on C, despite being identically zero on the topological concordance group.…”

“…Later, Levine [23] and Hedden [18] used the τ invariant from Heegaard Floer homology to show that the Mazur pattern and (n, 1) cables (n > 1) also do not induce homomorphisms. Most recently, A. Miller [34] used Casson-Gordon signatures to give an obstruction to patterns inducing a homomorphism even on the topological concordance group.…”