scholarly journals Tropically constructed Lagrangians in mirror quintic threefolds

2020 ◽  
Vol 8 ◽  
Author(s):  
Cheuk Yu Mak ◽  
Helge Ruddat
Keyword(s):  
Holomorphic Curves ◽  
Mapping Class ◽  
Rational Homology ◽  
Symplectic Mapping ◽  
Special Lagrangian ◽  
Tropical Curves ◽  
Tropical Curve ◽  
Rational Homology Spheres ◽  
Quintic Threefold ◽  
Degeneration Techniques

Abstract We use tropical curves and toric degeneration techniques to construct closed embedded Lagrangian rational homology spheres in a lot of Calabi-Yau threefolds. The homology spheres are mirror dual to the holomorphic curves contributing to the Gromov-Witten (GW) invariants. In view of Joyce’s conjecture, these Lagrangians are expected to have special Lagrangian representatives and hence solve a special Lagrangian enumerative problem in Calabi-Yau threefolds. We apply this construction to the tropical curves obtained from the 2,875 lines on the quintic Calabi-Yau threefold. Each admissible tropical curve gives a Lagrangian rational homology sphere in the corresponding mirror quintic threefold and the Joyce’s weight of each of these Lagrangians equals the multiplicity of the corresponding tropical curve. As applications, we show that disjoint curves give pairwise homologous but non-Hamiltonian isotopic Lagrangians and we check in an example that $>300$ mutually disjoint curves (and hence Lagrangians) arise. Dehn twists along these Lagrangians generate an abelian subgroup of the symplectic mapping class group with that rank.

Strongly-cyclic branched coverings and the Alexander polynomial of knots in rational homology spheres

2007 ◽  
Vol 142 (2) ◽  
pp. 259-268 ◽  
Author(s):  
YUYA KODA
Keyword(s):  
Fundamental Group ◽  
Homology Group ◽  
Alexander Polynomial ◽  
Homology Sphere ◽  
Cyclic Covering ◽  
Rational Homology ◽  
Branched Coverings ◽  
Branched Covering ◽  
Rational Homology Spheres ◽  
Rational Homology Sphere

AbstractLet K be a knot in a rational homology sphere M. In this paper we correlate the Alexander polynomial of K with a g-word cyclic presentation for the fundamental group of the strongly-cyclic covering of M branched over K. We also give a formula for the order of the first homology group of the strongly-cyclic branched covering.

scholarly journals ON ALEXANDER MODULES AND BLANCHFIELD FORMS OF NULL-HOMOLOGOUS KNOTS IN RATIONAL HOMOLOGY SPHERES

2012 ◽  
Vol 21 (05) ◽  
pp. 1250042 ◽  
Author(s):  
DELPHINE MOUSSARD
Keyword(s):  
Isomorphism Class ◽  
The Other ◽  
Rational Homology ◽  
Rational Homology Spheres

In this paper, we give a classification of Alexander modules of null-homologous knots in rational homology spheres. We characterize these modules [Formula: see text] equipped with their Blanchfield forms ϕ, and the modules [Formula: see text] such that there is a unique isomorphism class of [Formula: see text], and we prove that for the other modules [Formula: see text], there are infinitely many such classes. We realize all these [Formula: see text] by explicit knots in ℚ-spheres.

scholarly journals Refined curve counting with tropical geometry

2015 ◽  
Vol 152 (1) ◽  
pp. 115-151 ◽  
Author(s):  
Florian Block ◽  
Lothar Göttsche
Keyword(s):  
Tropical Geometry ◽  
Ruled Surfaces ◽  
Plane Curves ◽  
Toric Surfaces ◽  
Formula One ◽  
Tropical Curves ◽  
Tropical Curve ◽  
Severi Variety ◽  
Floor Diagrams

The Severi degree is the degree of the Severi variety parametrizing plane curves of degree $d$ with ${\it\delta}$ nodes. Recently, Göttsche and Shende gave two refinements of Severi degrees, polynomials in a variable $y$, which are conjecturally equal, for large $d$. At $y=1$, one of the refinements, the relative Severi degree, specializes to the (non-relative) Severi degree. We give a tropical description of the refined Severi degrees, in terms of a refined tropical curve count for all toric surfaces. We also refine the equivalent count of floor diagrams for Hirzebruch and rational ruled surfaces. Our description implies that, for fixed ${\it\delta}$, the refined Severi degrees are polynomials in $d$ and $y$, for large $d$. As a consequence, we show that, for ${\it\delta}\leqslant 10$ and all $d\geqslant {\it\delta}/2+1$, both refinements of Göttsche and Shende agree and equal our refined counts of tropical curves and floor diagrams.

scholarly journals LINEAR INDEPENDENCE IN THE RATIONAL HOMOLOGY COBORDISM GROUP

2019 ◽  
pp. 1-12 ◽  
Author(s):  
Marco Golla ◽  
Kyle Larson
Keyword(s):  
Infinite Order ◽  
Double Cover ◽  
Rational Homology ◽  
Cobordism Group ◽  
Connected Sums ◽  
Knot Concordance ◽  
Linearly Independent ◽  
Rational Homology Spheres ◽  
Homology Cobordism ◽  
Correction Terms

We give simple homological conditions for a rational homology 3-sphere $Y$ to have infinite order in the rational homology cobordism group $\unicode[STIX]{x1D6E9}_{\mathbb{Q}}^{3}$ , and for a collection of rational homology spheres to be linearly independent. These translate immediately to statements about knot concordance when $Y$ is the branched double cover of a knot, recovering some results of Livingston and Naik. The statements depend only on the homology groups of the 3-manifolds, but are proven through an analysis of correction terms and their behavior under connected sums.

scholarly journals Symplectic mapping class group relations generalizing the chain relation

2016 ◽  
Vol 27 (12) ◽  
pp. 1650096 ◽  
Author(s):  
Bahar Acu ◽  
Russell Avdek
Keyword(s):  
Mapping Class Group ◽  
Homogeneous Polynomial ◽  
Class Group ◽  
Symplectic Manifolds ◽  
Mapping Class ◽  
Symplectic Mapping ◽  
Group Relations ◽  
Type Boundary ◽  
Polynomial Formula ◽  
Classical Chain

In this paper, we examine mapping class group relations of some symplectic manifolds. For each [Formula: see text] and [Formula: see text], we show that the [Formula: see text]-dimensional Weinstein domain [Formula: see text], determined by the degree [Formula: see text] homogeneous polynomial [Formula: see text], has a Boothby–Wang type boundary and a right-handed fibered Dehn twist along the boundary that is symplectically isotopic to a product of right-handed Dehn twists along Lagrangian spheres. We also present explicit descriptions of the symplectomorphisms in the case [Formula: see text] recovering the classical chain relation for the torus with two boundary components.

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