Essential twisted surfaces in alternating link complements

TL;DR

This paper introduces twisted checkerboard surfaces in alternating link complements, which better capture the geometric properties of links with many crossings, and proves their essentiality in the link complement.

Contribution

It generalizes checkerboard surfaces to immersed twisted surfaces, demonstrating their essentiality and analyzing their properties in alternating link complements.

Findings

Twisted checkerboard surfaces are essential in alternating link complements.

These surfaces better reflect the geometry of links with many crossings.

The paper analyzes homotopy classes of arcs on these surfaces.

Abstract

Checkerboard surfaces in alternating link complements are used frequently to determine information about the link. However, when many crossings are added to a single twist region of a link diagram, the geometry of the link complement stabilizes (approaches a geometric limit), but a corresponding checkerboard surface increases in complexity with crossing number. In this paper, we generalize checkerboard surfaces to certain immersed surfaces, called twisted checkerboard surfaces, whose geometry better reflects that of the alternating link in many cases. We describe the surfaces, show that they are essential in the complement of an alternating link, and discuss their properties, including an analysis of homotopy classes of arcs on the surfaces in the link complement.