We show that there exists a Z ∞ -summand in the subgroup of the knot concordance group generated by knots with trivial Alexander polynomial. To this end we use the invariant Upsilon Υ recently introduced by Ozsváth, Stipsicz and Szabó using knot Floer homology. We partially compute Υ of (n, 1)-cable of the Whitehead double of the trefoil knot. For this computation of Υ, we determine a sufficient condition for two satellite knots to have identical Υ for any pattern with nonzero winding number.
We show that every good boundary link with a pair of derivative links on a Seifert surface satisfying a homotopically trivial plus assumption is freely slice. This subsumes all previously known methods for freely slicing good boundary links with two or more components, and provides new freely slice links.
We give infinitely many 2-component links with unknotted components which are topologically concordant to the Hopf link, but not smoothly concordant to any 2-component link with trivial Alexander polynomial. Our examples are pairwise non-concordant.
We give two infinite families of examples of closed, orientable, irreducible 3-manifolds M such that b1(M ) = 1 and π1(M ) has weight 1, but M is not the result of Dehn surgery along a knot in the 3-sphere. This answers a question of Aschenbrenner, Friedl and Wilton, and provides the first examples of irreducible manifolds with b1 = 1 that are known not to be surgery on a knot in the 3-sphere. One family consists of Seifert fibered 3-manifolds, while each member of the other family is not even homology cobordant to any Seifert fibered 3-manifold. None of our examples are homology cobordant to any manifold obtained by Dehn surgery along a knot in the 3-sphere.
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