The knot Floer complex and the smooth concordance group

Abstract

Abstract. We define a new smooth concordance homomorphism based on the knot Floer complex and an associated concordance invariant, ε. As an application, we show that an infinite family of topologically slice knots are independent in the smooth concordance group.

An infinite-rank summand of topologically slice knots

“…These knots were used in [Hom11a] to give an infinite rank subgroup of C T S , and Lemma 6.10 of that paper shows that a(K n ) = (1, n, . .…”

“…The {−1, 0, 1}-valued concordance invariant ε [Hom11a] is similarly defined in terms of the vanishing of certain natural maps on subquotient complexes. Let τ = τ (K).…”

“…In this section, we will recall and extend certain numerical invariants associated to [CF K ∞ (K)] from [Hom11a]. The invariant ε is defined by examining whether or not certain natural maps on subquotient complexes of CF K ∞ , corresponding to certain cobordisms, vanish on homology.…”

“…In [Hom11a], we showed that ε, while itself not a concordance homomorphism, can nonetheless be used to construct a highly nontrivial homomorphism from C to a non-Archimedean totally ordered group. We will use this construction to define infinitely many linearly independent Zvalued smooth concordance homomorphisms, which remain linearly independent when restricted to the topologically slice subgroup.…”

“…By Lemma 6.12 of [Hom11a], the knot D is ε-equivalent to T 2,3 . Thus, by [Hom11a, Proposition 4], we may consider CF K ∞ (T 2,3;n,n+1 ) instead of CF K ∞ (D n,n+1 ), where T 2,3;n,n+1 denotes the (n, n + 1)-cable of T 2,3 .…”