Oka and Alexander polynomials of symplectic curves and divisibility relations

TL;DR

The paper extends Libgober’s divisibility results for Alexander and Oka polynomials from complex plane curves to symplectic curves in CP^2, using a symplectic analogue of the pencil and braid-monodromy framework. It introduces a robust handle-decomposition approach for curve complements and develops both global and local divisibility statements: Δ_{C,L}|Δ_{K∞} and Δ^φ_{C,L}|Δ^φ_{K∞}, with refinements involving twisted and hatΔ polynomials and Euler characteristics of non-singular parts. The local theory leverages knot-theoretic tools (connected sums, knotifications, and Borromean knots) to bound Δ^φ_{C,L} in terms of singularity data, yielding explicit exponents depending on genus and branching, and recovering classical complex-curve results as a special case. Together, these results impose strong constraints on the possible Alexander and Oka polynomials for symplectic curves and hint at new avenues for understanding the topology of symplectic curve complements and their monodromies.

Abstract

We prove Libgober's divisibility relations for Oka and Alexander polynomials of symplectic curves in the complex projective plane. Along the way, we give new proofs of the divisibility relations for the Alexander polynomials of complex algebraic curves with respect to a generic line at infinity.

Figures (6)

• Figure 1: The Borromean knot $B_{2g}$. The dotted curves represent 1-handles (or 0-surgeries), see GompfStipsicz.
• Figure 2: The Borromean rings.
• Figure 3: A schematic example of a handle decomposition of the neighbourhood of a reducible curve: here the curve has two components $C_{\rm b}$ and $C_{\rm g}$, and we colour the components of the links of their singularities by blue or green according to which curve they belong to. In the left-most ball, we see a trefoil and the two blue arcs represent a connected sum with a (blue) Borromean knot $B_1$, while the other blue components are unknots, so $C_{\rm b}$ has genus 1 and one simple cusp. In the bottom-most ball we see a green Hopf link, and the arc connecting its two components represents their knotification, so $C_{\rm g}$ has genus 0 and one singularity which is a simple node. $C_{\rm b}$ and $C_{\rm g}$ intersect at two points, which can be seen from the two balls on the right. On the top one we see a Hopf link, so $C_{\rm b}$ and $C_{\rm g}$ are non-singular and intersect transversely at that point. On the bottom ball we see a triple Hopf link (three fibres of the Hopf fibration), two green and one blue: this means that $C_{\rm b}$ and $C_{\rm g}$ intersect at the node of $C_{\rm g}$, in a generic way. The other arc connect the various coloured links along the corresponding curves. In this case, the result is a 2-component link in $(S^1\times S^2)^{\#4}$.
• Figure 4: $\Gamma'$ and its neighbourhood $\nu(\Gamma')$ in $\mathbb{D}.$
• Figure 5: The decomposition of $(\mathbb{D}\times \mathbb{D}_y)\setminus \nu(\mathcal{C})$.

Theorems & Definitions (46)

• Theorem 1.1
• Theorem 1.2
• Corollary 1.3
• Remark 1.4
• Corollary 1.5
• Remark 1.6
• Remark 2.1
• Lemma 2.2
• proof
• Definition 2.3