On independence of iterated Whitehead doubles in the knot concordance group

Abstract

Let $D(K)$ be the positively-clasped untwisted Whitehead double of a knot $K$, and $T_{p,q}$ be the $(p,q)$ torus knot. We show that $D(T_{2,2m+1})$ and $D^2(T_{2,2m+1})$ are linearly independent in the smooth knot concordance group $\mathcal{C}$ for each $m\geq 2$. Further, $D(T_{2,5})$ and $D^2(T_{2,5})$ generate a $\mathbb{Z}\oplus\mathbb{Z}$ summand in the subgroup of $\mathcal{C}$ generated by topologically slice knots. We use the concordance invariant $δ$ of Manolescu and Owens, using Heegaard Floer correction term. Interestingly, these results are not easily shown using other concordance invariants such as the $τ$-invariant of knot Floer theory and the $s$-invariant of Khovanov homology. We also determine the infinity version of the knot Floer complex of $D(T_{2,2m+1})$ for any $m\geq 1$ generalizing a result for $T_{2,3}$ of Hedden, Kim and Livingston.