Connected Floer homology of covering involutions

Abstract

Using the covering involution on the double branched cover of S 3 branched along a knot, and adapting ideas of Hendricks-Manolescu and Hendricks-Hom-Lidman, we define new knot invariants and apply them to deduce novel linear independence results in the smooth concordance group of knots.

Knot Floer homology and surgery on equivariant knots

“…Hence, we have identified a covering action on HF `p´Σp2, 3, 7qq with the action of τ s K on A 0 . In fact, readers may explicitly compute the branching knot, identify it as a Montesinos knot and refer to [AKS20] to obtain that the action on HF `p´Σp2, 3, 7qq is exactly as in the right-hand side of Figure 1. Note however that, identifying the branching knot even in this simple case is not immediate, as the knot in question has 12 crossings.…”

“…In [DHM20] Dai-Hedden and the author studied several corks that can be obtained as surgery on a strongly invertible knot where the cork-twist involution corresponds to the induced action of the symmetry on the surgered manifold. In both the studies [AKS20] and [DHM20], the key tool was 1 to understand the induced action of the involution on the Heegaard Floer chain complex of the underlying 3-manifolds. On the other side of the coin, in [DMS22] Dai-Stoffrengen and the author showed the action of an involution on the knot Floer complex of an equivariant knot can be used to produce equivariant concordance invariants which bound equivariant 4-genus of equivariant knots, and it can be used to detect exotic slice disks.…”

Branched covers bounding rational homology balls

“…[16,20,29,33]) applied to the cyclic branched covers of K n all vanish. As mentioned above, the fact that the involution induced by the deck transformation on Σ 2 (K n ; Z) extends to a rational homology ball (in fact it is a Z 2 homology ball) implies that sliceness obstructions such as [3,8] vanish.…”

“…To put this question in a more algebraic framework, notice that Σ q (−K) = −Σ q (K) (where −K is the reverse of the mirror image of the knot K and −Y is the three-manifold Y with reversed orientation) and Σ q (K 1 #K 2 ) = Σ q (K 1 )#Σ q (K 2 ). Hence the map K → Σ q (K) descends to a homomorphism C → Θ 3 Q , where C denotes the smooth concordance group of knots in S 3 , and Θ 3 Q is the smooth rational homology cobordism group of rational homology spheres. We then let ϕ : C → q∈Q Θ 3 Q , be the homomorphism given by [K] → ([Σ q (K)]) q∈Q , and note that [K] ∈ ker ϕ exactly when all the prime power fold cyclic branched covers of K bound rational homology balls.…”

Strongly invertible knots, invariant surfaces, and the Atiyah-Singer signature theorem

“…Some work regarding the equivariant four-genus of periodic and strongly invertible knots has been done recently by Issa and the second author [BI21]. Other work regarding the four-dimensional topology of symmetric knots has been done by Davis and Naik [DN06], and by Dai, Hedden and Mallick [DHM20], building on the methods of [AKS20] based on ideas from involutive Heegaard Floer homology [HM17,HHL21]. In [Tro62] Trotter defines a bilinear pairing θ : H 1 (F ) ⊗ H 1 (F ) → Z associated to a Seifert surface F ⊂ S 3 .…”