Authors derive a closed-form formula for Gompf's θ-invariant of canonical contact structures on Seifert fibered 3-manifolds and a recursive formula for general plumbing trees using Legendrian surgery descriptions.
Heegaard Floer homology and maximal twisting numbers
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abstract
We adapt the Ozsv\'ath-Szab\'o full path algorithm to every star-shaped graph and establish a correspondence between negative-twisting tight contact structures on any Seifert fibred space over $S^2$, and its Heegaard Floer homology groups equipped with the Alexander filtration induced by the regular fibre. This provides the complete classification of negative-twisting structures on these manifolds; in particular, we distinguish them by their contact invariant $c^+$. We prove that every such structure is symplectically fillable and extend a known obstruction to Stein fillability. In addition, we show that the number of negative-twisting structures can be expressed combinatorially in terms of the Seifert coefficients of the star-shaped graph, while their $d_3$-invariant and homotopy type are determined explicitly through our correspondence. Our results also complete the classification of fillable structures on any small Seifert fibred space.
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Brieskorn spheres Σ(a1,...,an) obstruct rational homology ball symplectic fillings for any contact structure on -Y when n=3 or without half convex Giroux torsion for n>3, with limited exceptions for Milnor fillable cases, and those with vanishing correction terms have at most two fillable structures
Provides the complete list of Brieskorn spheres carrying at most two symplectically fillable contact structures up to isotopy.
citing papers explorer
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The Gompf $\theta$-Invariant of Canonical Contact Structures via Legendrian Surgery
Authors derive a closed-form formula for Gompf's θ-invariant of canonical contact structures on Seifert fibered 3-manifolds and a recursive formula for general plumbing trees using Legendrian surgery descriptions.
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Brieskorn spheres and rational homology ball symplectic fillings
Brieskorn spheres Σ(a1,...,an) obstruct rational homology ball symplectic fillings for any contact structure on -Y when n=3 or without half convex Giroux torsion for n>3, with limited exceptions for Milnor fillable cases, and those with vanishing correction terms have at most two fillable structures
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Brieskorn spheres with two fillable contact structures
Provides the complete list of Brieskorn spheres carrying at most two symplectically fillable contact structures up to isotopy.