Linear independence of cables in the knot concordance group

Abstract

We produce infinite families of knots { K i } i ≥ 1 \{K^i\}_{i\ge 1} for which the set of cables { K p , 1 i } i , p ≥ 1 \{K^i_{p,1}\}_{i,p\ge 1} is linearly independent in the knot concordance group, C \mathcal {C} . We arrange that these examples lie arbitrarily deep in the solvable and bipolar filtrations of C \mathcal {C} , denoted by { F n } \{\mathcal {F}_n\} and { B n } \{\mathcal {B}_n\} respectively. As a consequence, this result cannot be reached by any combination of algebraic concordance invariants, Casson-Gordon invariants, and Heegaard-Floer invariants such as τ \tau , ε \varepsilon , and Υ \Upsilon . We give two applications of this result. First, for any n ≥ 0 n\ge 0 , there exists an infinite family { K i } i ≥ 1 \{K^i\}_{i\geq 1} such that for each fixed i i , { K 2 j , 1 i } j ≥ 0 \{K^i_{2^j,1}\}_{j\geq 0} is a basis for an infinite rank summand of F n \mathcal {F}_n and { K p , 1 i } i , p ≥ 1 \{K^i_{p,1}\}_{i, p\geq 1} is linearly independent in F n / F n .5 \mathcal {F}_{n}/\mathcal {F}_{n.5} . Second, for any n ≥ 1 n\ge 1 , we give filtered counterexamples to Kauffman’s conjecture on slice knots by constructing smoothly slice knots with genus one Seifert surfaces where one derivative curve has nontrivial Arf invariant and the other is nontrivial in both F n / F n .5 \mathcal {F}_n/\mathcal {F}_{n.5} and B n − 1 / B n + 1 \mathcal {B}_{n-1}/\mathcal {B}_{n+1} . We also give examples of smoothly slice knots with genus one Seifert surfaces such that one derivative has nontrivial Arf invariant and the other is topologically slice but not smoothly slice.

Torsion in the knot concordance group and cabling

“…There are many nonslice knots where their (p, 1)-cables have infinite order in C. For example, if K has nontrivial signature function [Tri69,Lev69], then a formula of Litherland [Lit79] implies that K p,1 has infinite order in C. In fact, the same formula implies that the set of cables {K p,1 } p>1 is linearly independent in C. A similar conclusions obtained by various different techniques can be found in [KP18,FPR19,Che21,DPR21]. These results crucially use the fact that the starting knot K has infinite order in C (e.g.…”

Slice knots and knot concordance

“…For other related work, see also [Jia81,Liv99,DPR21]. By contrast, little is known about the "other half" of the filtration.…”

“…For now, we note that the examples from [COT03, COT04, CHL09, CHL11a] all have genus one. The examples from [DPR21] are cables of genus one knots, and have large Seifert genus and smooth slice genus, but may well have topological slice genus one. In general it is difficult to bound the topological slice genus of knots lying deep in the solvable filtration.…”