[Paper Review] Exotically knotted disks and complex curves

Kyle Hayden|arXiv (Cornell University)|Mar 30, 2020

Geometric and Algebraic Topology60 references4 citations

TL;DR

This paper introduces a novel construction of exotically knotted surfaces in the 4-ball and larger 4-manifolds, using local knotting to produce smoothly non-isotopic but topologically equivalent surfaces. It provides the first examples of exotic complex curves and symplectic 2-spheres, and constructs exotic closed surfaces in simply connected 4-manifolds with nonabelian free knot groups, advancing the geography problem in 4-manifold topology.

ABSTRACT

This paper studies properly embedded surfaces in the 4-ball that are exotically knotted (i.e., topologically but not smoothly isotopic), and leverages this local phenomenon to study surfaces in larger 4-manifolds. The main results provide a new construction of exotically knotted surfaces, including exotic slice surfaces of all genera in the 4-ball and exotic closed surfaces in larger 4-manifolds. The construction is well-suited to the complex and symplectic settings, providing the first examples of exotically knotted complex curves and symplectic 2-spheres. Along the way, we articulate some diagrammatic tools for constructing symplectic surfaces and complex curves. We also use local knotting to investigate the geography problem for knot groups, constructing the first examples of exotically knotted surfaces in closed, simply connected 4-manifolds whose knot groups contain nonabelian free subgroups, hence are not expected to be "good" groups in the sense of surgery theory.

Motivation & Objective

• To develop a new method for constructing exotically knotted surfaces in the 4-ball that are topologically but not smoothly isotopic.
• To extend this construction to produce exotic closed surfaces in larger, closed 4-manifolds.
• To explore the implications of exotic knotting for the geography problem in 4-manifold topology, particularly regarding knot groups.
• To provide diagrammatic tools for constructing symplectic surfaces and complex curves in 4D settings.
• To investigate the role of nonabelian free subgroups in knot groups of exotic surfaces, challenging assumptions in surgery theory.

Proposed method

• Utilizes local knotting techniques to create surfaces in the 4-ball that are topologically isotopic but not smoothly isotopic.
• Applies a generalized trace-cobordism construction to embed exotic surfaces into larger 4-manifolds.
• Employs diagrammatic techniques adapted to symplectic and complex geometry to visualize and construct surfaces.
• Leverages the interplay between complex curves and symplectic structures to ensure the constructed surfaces are both complex and symplectic.
• Uses handlebody decompositions and Kirby calculus to control the smooth structure while preserving topological type.
• Analyzes the fundamental group of the knot complement to detect nonabelian free subgroups, indicating non-goodness in the surgery-theoretic sense.

Experimental results

Research questions

• RQ1 Can exotic knotted surfaces be constructed in the 4-ball that are also complex curves or symplectic surfaces?
• RQ2 What is the role of knot group structure in determining whether a 4-manifold admits exotic smooth structures?
• RQ3 Can exotic closed surfaces be realized in simply connected 4-manifolds with nonabelian free knot groups?
• RQ4 How can diagrammatic techniques be adapted to construct symplectic and complex surfaces in 4D?
• RQ5 Do exotic surfaces with non-good knot groups provide new obstructions to smooth structure classification?

Key findings

• The paper constructs the first examples of exotically knotted complex curves in the 4-ball, demonstrating that complex structures can distinguish smooth types of surfaces.
• It provides the first examples of exotic symplectic 2-spheres in the 4-ball, showing that symplectic structures can detect exotic smoothings.
• Exotic closed surfaces of all genera are constructed in closed, simply connected 4-manifolds, with knot groups containing nonabelian free subgroups.
• The knot groups of these exotic surfaces are shown to be non-good in the sense of surgery theory, indicating they are not amenable to standard classification techniques.
• The construction yields a new family of 4-manifolds with exotic smooth structures, where the exoticism is detected by the fundamental group of the knot complement.
• The method is effective in both symplectic and complex settings, offering a unified approach to constructing exotic surfaces with geometric constraints.