Every genus one algebraically slice knot is 1-solvable

Abstract

Cochran, Orr, and Teichner developed a filtration of the knot concordance group indexed by half integers called the solvable filtration. Its terms are denoted by F n . It has been shown that F n /F n.5 is a very large group for n ≥ 0. For a generalization to the setting of links the third author showed that F n.5 /F n+1 is non-trivial. In this paper we provide evidence that for knots F 0.5 = F 1 . In particular we prove that every genus 1 algebraically slice knot is 1-solvable.Proposition ([COT03], Theorem 8.9). If L is a derivative for K and L is n-solvable then K is (n + 1)-solvable. If L is n.5-solvable, then K is (n + 1.5)-solvable.

Primary decomposition in the smooth concordance group of topologically slice knots

“…For integers n > 0, it is unknown whether F n.5 /F n+1 is nontrivial. Recently Davis, Martin, Otto and Park showed that elements in F 0.5 represented by a genus one knot are contained in F 1 [19]. For the other half of the associated graded F n /F n.5 , Cochran, Harvey and Leidy provided strong evidences which support the conjecture that the associated graded F n /F n.5 is right primary decomposable, for all integers n ≥ 0 [13,14].…”

Slice knots and knot concordance

“…For some n ∈ N, is the group F n.5 / F n+1 nontrivial? For example, it was shown in [DMOP19] that genus one knots which are 0.5-solvable (equivalently, algebraically slice) are also 1-solvable. We note that there is an analogue of the solvable filtration for m-component (string) links, denoted by {F…”

Homomorphism obstructions for satellite maps

“…For any knot K, the knot P 0 (K) is genus 1 and algebraically slice, hence by [DMOP19] is 1-solvable. Therefore, Theorem 4.4 implies that P (K) = (P n • • • • • P 1 )(P 0 (K)) is (n + 1)-solvable, and we have established condition (2).…”