scholarly journals Genera of knots in the complex projective plane

2020 ◽  
Vol 29 (12) ◽  
pp. 2050081
Author(s):  
Jake Pichelmeyer
Keyword(s):  
Projective Plane ◽  
Lower Bounds ◽  
Infinite Family ◽  
Upper Bounds ◽  
Complex Projective Plane ◽  
Low Dimensional Topology ◽  
Low Dimensional ◽  
Complex Projective

Our goal is to systematically compute the [Formula: see text]-genus of as many prime knots up to 8-crossings as possible. We obtain upper bounds on the [Formula: see text]-genus via coherent band surgery. We obtain lower bounds by obstructing homological degrees of potential slice discs. The obstructions are pulled from a variety of sources in low-dimensional topology and adapted to [Formula: see text]. There are 27 prime knots and distinct mirrors up to 7-crossings. We now know the [Formula: see text]-genus of all of these knots. There are 64 prime knots and distinct mirrors up to 8-crossings. We now know the [Formula: see text]-genus of all but 6 of these knots, where the [Formula: see text]-genus was not determined explicitly, it was narrowed down to 2 possibilities. As a consequence of this work, we show an infinite family of knots such that the [Formula: see text]-genus of each knot differs from that of its mirror.

scholarly journals SMOOTH, NONSYMPLECTIC EMBEDDINGS OF RATIONAL BALLS IN THE COMPLEX PROJECTIVE PLANE

2020 ◽  
Vol 71 (3) ◽  
pp. 997-1007
Author(s):  
Brendan Owens
Keyword(s):  
Projective Plane ◽  
Infinite Family ◽  
Complex Projective Plane ◽  
Rational Homology ◽  
Complex Projective

Abstract We exhibit an infinite family of rational homology balls, which embed smoothly but not symplectically in the complex projective plane. We also obtain a new lattice embedding obstruction from Donaldson’s diagonalization theorem and use this to show that no two of our examples may be embedded disjointly.

scholarly journals On 3-syzygy and unexpected plane curves

2021 ◽  
Author(s):  
Grzegorz Malara ◽  
Piotr Pokora ◽  
Halszka Tutaj-Gasińska
Keyword(s):  
Projective Plane ◽  
Plane Curves ◽  
Complex Projective Plane ◽  
Geometric Properties ◽  
Complex Projective

AbstractIn this note we study curves (arrangements) in the complex projective plane which can be considered as generalizations of free curves. We construct families of arrangements which are nearly free and possess interesting geometric properties. More generally, we study 3-syzygy curve arrangements and we present examples that admit unexpected curves.

Spectral Sequences

2020 ◽  
pp. 45-56
Author(s):  
Loring W. Tu
Keyword(s):  
Projective Plane ◽  
Spectral Sequence ◽  
Fiber Bundles ◽  
Spectral Sequences ◽  
Complex Projective Plane ◽  
Computational Tool ◽  
Short Introduction ◽  
Complex Projective

This chapter focuses on spectral sequences. The spectral sequence is a powerful computational tool in the theory of fiber bundles. First introduced by Jean Leray in the 1940s, it was further refined by Jean-Louis Koszul, Henri Cartan, Jean-Pierre Serre, and many others. The chapter provides a short introduction, without proofs, to spectral sequences. As an example, it computes the cohomology of the complex projective plane. The chapter then details Leray's theorem. A spectral sequence is like a book with many pages. Each time one turns a page, one obtains a new page that is the cohomology of the previous page.

The Space of Harmonic Maps from the 2-Sphere to the Complex Projective Plane

1997 ◽  
Vol 40 (3) ◽  
pp. 285-295 ◽  
Author(s):  
T. Arleigh Crawford
Keyword(s):  
Projective Plane ◽  
Harmonic Maps ◽  
Complex Manifolds ◽  
Complex Projective Plane ◽  
Fixed Degree ◽  
Complex Projective ◽  
Path Connected

AbstractIn this paper we study the topology of the space of harmonic maps from S2 to ℂℙ2.We prove that the subspaces consisting of maps of a fixed degree and energy are path connected. By a result of Guest and Ohnita it follows that the same is true for the space of harmonic maps to ℂℙn for n ≥ 2. We show that the components of maps to ℂℙ2 are complex manifolds.

scholarly journals Real Hypersurfaces of Nonflat Complex Projective Planes Whose Jacobi Structure Operator Satisfies a Generalized Commutative Condition

2016 ◽  
Vol 2016 ◽  
pp. 1-6
Author(s):  
Theocharis Theofanidis
Keyword(s):  
Projective Plane ◽  
Projective Planes ◽  
Additional Restriction ◽  
Real Hypersurfaces ◽  
Specific Function ◽  
Complex Projective Plane ◽  
Jacobi Structure ◽  
Complex Projective

Real hypersurfaces satisfying the conditionϕl=lϕ(l=R(·,ξ)ξ)have been studied by many authors under at least one more condition, since the class of these hypersurfaces is quite tough to be classified. The aim of the present paper is the classification of real hypersurfaces in complex projective planeCP2satisfying a generalization ofϕl=lϕunder an additional restriction on a specific function.

ON FINITE BRANCHED UNIFORMIZATIONS OF THE PROJECTIVE PLANE

2013 ◽  
Vol 24 (02) ◽  
pp. 1350017
Author(s):  
A. MUHAMMED ULUDAĞ ◽  
CELAL CEM SARIOĞLU
Keyword(s):  
Projective Plane ◽  
Fundamental Group ◽  
Free Product ◽  
Plane Curve ◽  
Complex Projective Plane ◽  
Galois Covering ◽  
Cyclic Groups ◽  
Galois Coverings ◽  
Complex Projective ◽  
Branching Index

We give a brief survey of the so-called Fenchel's problem for the projective plane, that is the problem of existence of finite Galois coverings of the complex projective plane branched along a given divisor and prove the following result: Let p, q be two integers greater than 1 and C be an irreducible plane curve. If there is a surjection of the fundamental group of the complement of C into a free product of cyclic groups of orders p and q, then there is a finite Galois covering of the projective plane branched along C with any given branching index.

scholarly journals Topological Realizations of Line Arrangements

2017 ◽  
Vol 2019 (8) ◽  
pp. 2295-2331
Author(s):  
Daniel Ruberman ◽  
Laura Starkston
Keyword(s):  
Finite Field ◽  
Projective Plane ◽  
Contact Structure ◽  
Complex Projective Plane ◽  
Real Projective Plane ◽  
Incidence Relation ◽  
Combinatorial Configuration ◽  
Important Variation ◽  
Graph Manifolds ◽  
Complex Projective

Abstract A venerable problem in combinatorics and geometry asks whether a given incidence relation may be realized by a configuration of points and lines. The classic version of this would ask for lines in a projective plane over a field. An important variation allows for pseudolines: embedded circles (isotopic to $\mathbb R\rm{P}^1$) in the real projective plane. In this article we investigate whether a configuration is realized by a collection of 2-spheres embedded, in symplectic, smooth, and topological categories, in the complex projective plane. We find obstructions to the existence of topologically locally flat spheres realizing a configuration, and show for instance that the combinatorial configuration corresponding to the projective plane over any finite field is not realized. Such obstructions are used to show that a particular contact structure on certain graph manifolds is not (strongly) symplectically fillable. We also show that a configuration of real pseudolines can be complexified to give a configuration of smooth, indeed symplectically embedded, 2-spheres.

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