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 $m\ge 2$. 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 $\delta$ of Manolescu and Owens, using Heegaard Floer correction term. Interestingly, these results are not easily shown using other concordance invariants such as the $\tau$-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 $m\ge 1$ generalizing a result for $T_{2,3}$ of Hedden, Kim and Livingston.

14 pages, 5 figures, errors fixed, the version to appear in JKTR

References in corpus (3)

• Concordance homomorphisms from knot Floer homology
• Filtering smooth concordance classes of topologically slice knots
• Heegaard Floer homology and rational cuspidal curves

Cited by in corpus (2)

• Satellites of Infinite Rank in the Smooth Concordance Group
• Instantons, special cycles, and knot concordance