More concordance homomorphisms from knot Floer homology

Abstract

We define an infinite family of linearly independent, integer-valued smooth concordance homomorphisms. Our homomorphisms are explicitly computable and rely on local equivalence classes of knot Floer complexes over the ring F[U, V ]/(U V = 0). We compare our invariants to other concordance homomorphisms coming from knot Floer homology, and discuss applications to topologically slice knots, concordance genus, and concordance unknotting number.

Ribbon knots, cabling, and handle decompositions

“…Note that the notion of reducedness is preserved under basis changes of the form x i → x i + j∈J U j V m j x i j . We will use the grading conventions of [Zem19b]; see also [DHST19,Section 2]. Recall that multiplication by U lowers the Alexander grading by 1, multiplication by V raises the Alexander grading by 1, and the differential preserves the Alexander grading.…”

Linear independence of rationally slice knots

“…If K is rationally slice, then both ±1 surgeries on K bound rational homology balls. Hence the τ -invariant [HW16], and ϕ jinvariants [DHST19] all vanish for K and its mirror. (On the other hand, it is not known if…”

“…We refer to the pair (CFK F[U ,V] (K), ι K ) as the ι K -complex of a knot K. Moreover, Zemke [Zem19b] (see also [Zem19a, Theorem 1.5]) showed that up to an algebraic equivalence called local equivalence, the ι K -complex of a knot is a concordance invariant. We use a slightly coarser equivalence relation called almost local equivalence, which is motivated by [DHST18] (see also [DHST19]), to show the following. Note that the following theorem implies Theorem 1.1 immediately since the (p, −1)-cable of a rationally slice knot is rationally slice.…”

“…It can sometimes be cumbersome to work with ι K -complexes. Let (U, V) denote the ideal generated by U and V. Motivated by [DHST18] (see also [DHST19]), we make the following more relaxed definition:…”

An infinite-rank summand from iterated Mazur pattern satellite knots

“…Intuitively speaking, general satellite operations are quite different from the connected sum and hence often disregard the group structure of C. This feature has been exploited widely to construct linearly independent knots in various context (e.g. [4,5,7,12,20]). It is hence reasonable to expect Theorem 1.2 to be true, and it is even tempted to believe general iterated satellite knots are independent.…”