On Crossing Ball Structure in Knot and Link Complements

TL;DR

This paper introduces a new diagrammatic method for analyzing surfaces in knot and link complements, providing criteria for incompressibility and bounds on surface complexity.

Contribution

It develops a word-based mechanism for diagram analysis and establishes a finiteness theorem with bounds on Euler characteristics of embedded surfaces.

Findings

Provides a necessary condition for incompressible surfaces in knot/link complements.

Establishes a finiteness theorem for such surfaces.

Derives an upper bound on the Euler characteristic of these surfaces.

Abstract

We develop a word mechanism applied in knot and link diagrams for the illustration of a diagrammatic property. We also give a necessary condition for determining incompressible and pairwise incompressible surfaces, that are embedded in knot or link complements. Finally, we give a finiteness theorem and an upper bound on the Euler characteristic of such surfaces.