An infinite-rank summand of knots with trivial Alexander polynomial

TL;DR

This paper demonstrates the existence of an infinite-rank summand within the subgroup of the knot concordance group generated by knots with trivial Alexander polynomial, utilizing the Upsilon invariant from knot Floer homology.

Contribution

It establishes the presence of a $ ext{Z}^ ext{infinity}$-summand in the subgroup generated by trivial Alexander polynomial knots, using advanced invariants and partial computations.

Findings

Existence of an infinite-rank summand in the subgroup

Partial computation of Upsilon for specific satellite knots

A criterion for satellite knots to have identical Upsilon invariants

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.