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 ${\rm Y}$ recently introduced by Ozsváth, Stipsicz and Szabó using knot Floer homology. We partially compute ${\rm Y}$ of $(n,1)$-cable of the Whitehead double of the trefoil knot. For this computation of ${\rm Y}$, we determine a sufficient condition for two satellite knots to have identical ${\rm Y}$ for any pattern with nonzero winding number.

16 pages, 3 figures, Any questions or comments are welcome