Item

Abstract

We produce infinite families of knots $\{K^i\}_{i\geq 1}$ for which the set
of cables $\{K^i_{p,1}\}_{i,p\geq 1}$ is linearly independent in the knot
concordance group. We arrange that these examples lie arbitrarily deep in the
solvable and bipolar filtrations of the knot concordance group, denoted by
$\{F_n\}$ and $\{B_n\}$ 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 tau, epsilon, and Upsilon. We
give two applications of this result. First, for any n>=0, there exists an
infinite family $\{K^i\}_{i\geq 1}$ such that for each fixed i,
$\{K^i_{2^j,1}\}_{j\geq 0}$ is a basis for an infinite rank summand of $F_n$
and $\{K^i_{p,1}\}_{i, p\geq 1}$ is linearly independent in $F_{n}/F_{n.5}$.
Second, for any n>=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 $F_n/F_{n.5}$ and $B_{n-1}/B_{n+1}$. 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.