Involutive Heegaard Floer homology and rational cuspidal curves

Abstract

We use invariants of Hendricks and Manolescu coming from involutive Heegaard Floer theory to find constraints on possible configurations of singular points of a rational cuspidal curve of odd degree in the projective plane. We show that the results do not carry over to rational cuspidal curves of even degree.Theorem 4.10. Let C be a rational cuspidal curve of odd degree with two singular points z 1 and z 2 . Let K 1 and K 2 be links of singularities of z 1 and z 2 . Then at least one of K 1 and K 2 is an even L-space knot.We illustrate the above application by a simple example. It is a well-known result (see [18, Section 6.1.3]) but we give the first topological proof.Example 1.3. A rational cuspidal curve of degree 5 cannot have two singular points with Puiseux sequences (2; 11) and (2; 3). It also cannot have two singular points with Puiseux sequences (2; 7) and (2; 7).A rational cuspidal curve with Puiseux sequences (2; 9) and (2; 5) actually exists. It is item 7 in the list [19, Theorem 2.3.10]. See also [18, Section 6.1.3].This degree 5 example is quite remarkable from the following point of view. In [3] Bodnár and Némethi noted that the criterion of [4] does not actually restrict singular points, but only so-called multiplicity sequences. We do not give all the details, but point out that the criterion of [4] is unable to distinguish the case of singular points (2; 11), (2; 3) and (2; 9), (2; 5). We give more examples in Section 5. 1.3. Outline of the proof The main idea of the proof comes from [4], although technical problems already appear at an early stage. Consider a rational cuspidal curve C ⊂ CP 2 and let N be a tubular neighborhood of C. Let M = ∂N and set W = CP 2 \ N . As in [4] we identify M with a surgery on the connected sum of links of singularities of C. Moreover H k (W ; Q) = 0 for k > 0. The latter fact implies by [22] that for any Spin c structure s on M that extends to W we have d(M, s) = 0, where d is the Ozsváth-Szabó d-invariant. The equality d(M, s) = 0 was exploited in [4].In the present article we rely on a result of Hendricks and Manolescu, that for any Spin structure s on M that extends over W we have d(M, s) = d(M, s) = 0, where d and d are the invariants defined in [14]; see Section 2.1. We look at the canonical Spin structure on M , that is, the one corresponding to m = 0; see Section 2.2 for notation. The problem is that the canonical Spin structure extends over W if and only if deg C is odd. Therefore our results are restricted to curves of odd degree; see Section 4.1.If C has one singular point, then M is an L-space and it follows from [14, Section 4.4] that d(M, s) = d(M, s) = d(M, s). In particular, our result says nothing new for rational cuspidal curves with one singular point. However, if C has more than one singular point, the condition d(M, s) = d(M, s) becomes restrictive. The second, and actually, more difficult, part of the paper translates the equality d(M, s) = d(M, s) into a tractable condition on semigroups of singular points of C.Throughout, we let F = Z/2Z. In...

Involutive Heegaard Floer Homology and Plumbed Three-Manifolds

“…, r n be sequences of rational numbers, all differing from one another by even integers, such that (1) h 1 ą h 2 ą¨¨¨ą h n , (2) r 1 ă r 2 㨨¨ă r n , and (3) h n ě r n . 3 Recall from Section 2.3 that reflecting the graded root R k rσ`2s across the horizontal line of gradinǵ 1 yields a downwards opening graded root whose lattice homology computes the Heegaard Floer homology HF`p´Y, sq. Theorem 3.2 of [5] states that the degree of the lowermost J0-invariant vertex in this reflected root is 2μpY, sq.…”

“…Just as Heegaard Floer homology is more amenable to computations than Seiberg-Witten theory, involutive Heegaard Floer homology should be easier to calculate than its gauge-theoretic counterpart. So far, HFI • was computed in [8] for large surgeries on L-space and thin knots (including many hyperbolic examples); see also [3] for related applications. Moreover, a formula for HFI • of connected sums was established in [9].…”

“…are linearly independent in 3 Z , forming a Z ∞ subgroup. (The same manifolds were considered by Stoffregen in [24].…”

Connected sums and involutive knot Floer homology

“…Before we prove the above relations, we will show that they are sufficient to show that f = g. Equations (30) and (31) imply that A | V and B | V are linearly independent and hence form a basis of the two-dimensional vector space V ∨ = Hom F2 (V, F 2 ). Equations (32) and (33) then imply that their values on f (1) and g (1) agree, so f = g.…”

Heegaard Floer Homologies and Rational Cuspidal Curves. Lecture notes.

“…The proof involves involutive Floer theory as developed by Hendricks and Manolescu [27], which is beyond the scope of the present article. See [6] for details.…”