Concordance invariants from higher order covers

Abstract

We generalize the Manolescu-Owens smooth concordance invariant δ(K) of knots K ⊂ S 3 to invariants δ p n (K) obtained by considering covers of order p n , with p a prime. Our main result shows that for any prime p = 2, the thus obtained homomorphism ⊕ n∈N δ p n from the smooth concordance group to Z ∞ has infinite rank. We also show that unlike δ, these new invariants typically are not multiples of the knot signature, even for alternating knots. A significant portion of the article is devoted to exploring examples.

Rasmussen s-invariants of satellites do not detect slice knots

“…We remark that our argument for the s-invariant applies to the case of the τ -invariant, the ǫ-invariant and the knot Floer chain complex invariant [CF K ∞ (K)] of Hom [Hom14a,Hom14b], and the δ p k -invariant of Manolescu-Owens [MO07] and Jabuka [Jab12] as well. Using this, we observe that these invariants of arbitrary satellites, even when considered all together, do not detect slice knots:…”

Instantons, concordance, and Whitehead doubling

“…Indeed, by considering prime power order cyclic branched covers, we obtain an infinite family of homomorphisms Σ p k : C → Θ Z/p , where Σ p k denotes the map which takes the concordance class of a knot to the Z/phomology cobordism class of its p k -fold branched cyclic cover (here, p is a prime). One can obtain homomorphisms, δ p k : C → Θ Z/p → Z by considering the correction term of a particular Spin c structure on these covers [34,22]. Although we have not calculated these invariants (except in the case δ 2 , where they were calculated in [34]), we have used cobordism arguments similar to those presented here to obtain bounds which tightly constrain the behavior of δ p k (D(T r,s )).…”

Heegaard Floer homology and several families of Brieskorn spheres

“…2.7 The concordance invariant δ p n Let K ⊂ S 3 be a knot, let p be a prime, and let n ∈ N. Then let Σ p n (K) denote the p n -fold branched cover of S 3 branched along the knot K. Σ p n (K) is a rational homology sphere, and we let s 0 ∈ Spin c (Σ p n (K)) denote the element induced by the unique spin-structure. Manolescu and Owens [MO07] (for p n = 2) and Jabuka [Jab08] (for general p n ) define…”

“…Then HF + (M ) is as characterized by Table 1. Let p be a prime and K ⊂ S 3 be a knot. Using a particular d-invariant for the p n -fold cover of S 3 branched along K, Manolescu and Owens [MO07] (for p n = 2) and Jabuka [Jab08] (for any prime p and any n ∈ Z) define a concordance invariant δ p n Z; see §2.7 for the definition of this invariant. Theorem 2 provides some new δ p n -invariants for torus knots (although a few of the examples below appeared in [Jab08]).…”