An infinite-rank summand of knots with trivial Alexander polynomial

Abstract

We show that there exists a $\mathbb{Z}^\infty$-summand in the subgroup of the knot concordance group generated by knots with trivial Alexander polynomial. To this end we use the invariant Upsilon $\Upsilon$ recently introduced by Ozsv\'ath, Stipsicz and Szab\'o using knot Floer homology. We partially compute $\Upsilon$ of $(n,1)$-cable of the Whitehead double of the trefoil knot. For this computation of $\Upsilon$, we determine a sufficient condition for two satellite knots to have identical $\Upsilon$ for any pattern with nonzero winding number.