The Number of Closed Essential Surfaces in Montesinos Knots with Four Rational Tangles

TL;DR

This paper counts the number of closed essential surfaces of fixed genus in the complements of hyperbolic Montesinos knots with four rational tangles, revealing a precise, genus-dependent enumeration independent of crossing number.

Contribution

It provides an explicit count of closed essential surfaces in these knot complements, showing the number depends on genus and is independent of crossings, extending understanding of their topology.

Findings

Exactly 12 genus 2 surfaces exist in these complements.

Number of higher genus surfaces is 8 times Euler totient of g-1.

Count applies uniformly across all such hyperbolic Montesinos knots.

Abstract

In the complement of a hyperbolic Montesinos knot with 4 rational tangles, we investigate the number of closed, connected, essential, orientable surfaces of a fixed genus $g$, up to isotopy. We show that there are exactly 12 genus 2 surfaces and $8\phi(g - 1)$ surfaces of genus greater than 2, where $\phi(g - 1)$ is the Euler totient function of $g - 1$. Observe that this count is independent of the number of crossings of the knot. Moreover, this class of knots form an infinite class of hyperbolic 3-manifolds and the result applies to all such knot complements.