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

Brannon Basilio

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arXiv · 2204.01789

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

Brannon Basilio

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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.

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BibTeX RIS

Brannon Basilio. 2022-04-04. The Number of Closed Essential Surfaces in Montesinos Knots with Four Rational Tangles. https://arxiv.org/abs/2204.01789

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