Torsion in the knot concordance group and cabling

Abstract

We define a nontrivial mod 2 valued additive concordance invariant defined on the torsion subgroup of the knot concordance group using involutive knot Floer package. For knots not contained in its kernel, we prove that their iterated (odd, 1)-cables have infinite order in the concordance group and, among them, infinitely many are linearly independent. Furthermore, by taking (2, 1)-cables of the aforementioned knots, we present an infinite family of knots which are strongly rationally slice but not slice. 2020 Mathematics Subject Classification. 57K10, 57K18. Key words and phrases. Concordance knots, cabling, Horizontal almost ι K -complex. 1 One way to prove this is to use the cabling formulae for the Ozsváth-Szabó concordance invariant τ [OS03] and the Ozsváth-Stipsicz-Szabó concordance invariant Υ [OSS17]; see e.g. [Hom14a, Theorem 1] for τ and [FPR19, Proposition 5.3] for Υ.

Linear independence of rationally slice knots

“…Therefore K n has infinite order in the smooth concordance group. To prove the linear independence part of Theorem 1.1, we recall more facts about I U K from [KP22]. Recall from [KP22, Section 4] that, for each n ≥ 2, the horizontal almost ι K -complex C n is generated by elements a n , b n , c n , d n , x n , where a n and x n have bidegree (0, 0).…”

“…Remark 4.5. The argument used in the second part of the proof of Theorem 1.1 can be summarized as follows; note that the same argument was also used in [KP22]. Let {K n } be a sequence of rationally slice knots such that for any i ≥ 2, there exists a positive integer n i such that [K ni ] ≥ [C i ] in I U K .…”

“…The question of distinguishing strongly rationally slice knots from slice knots is very subtle; the first example of such a knot was found in [KP22]. By directly applying the proof of [KP22, Theorem 1.4] to the knots in Theorem 1.1, we immediately get the following corollary.…”

“…(3) We combine the result from (2) with results from [KP22] to compare our knots K n to certain standard complexes C n , with respect to the partial order on the horizontal almost ι K -local equivalence group (described in Section 2). Using properties of how the C n interact with the partial order, we arrive at the desired linear independence result.…”