Essential Surfaces in the Exterior of K13n586

TL;DR

This paper counts the isotopy classes of essential surfaces in the exterior of knot K13n586, revealing a novel connection between surface counts and the Euler totient function, and introduces a new method for analyzing surface components.

Contribution

It establishes the first known case where the number of essential surfaces by genus is explicitly determined, using a novel approach involving orbit counting under bijections.

Findings

Number of surfaces by genus equals Euler totient function.

First manifold with known surface counts for any genus.

Introduces a method to count surface components via orbit analysis.

Abstract

We count the number of isotopy classes of closed, connected, orientable, essential surfaces embedded in the exterior B of the knot K13n586.The main result is that the count of surfaces by genus is equal to the Euler totent function. This is the first manifold for which we know the number of surfaces for any genus. The main argument is to show when normal surfaces in B are connected by counting their number of components. We implement tools from Agol, Hass and Thurston to convert the problem of counting components of surfaces into counting the number of orbits in a set of integers under a collection of bijections defined on its subsets.