[Paper Review] Closed essential surfaces in the complements of large volume Berge knots

Kenneth L. Baker|ArXiv.org|Sep 4, 2005

Geometric and Algebraic Topology6 references3 citations

TL;DR

This paper develops an algorithm to classify all closed essential surfaces in the complements of large-volume Berge knots—specifically those lying on the fiber of a trefoil or figure-eight knot—using continued fraction expansions and twisted surface theory in once-punctured torus bundles. The key result shows that such knots can have arbitrarily many distinct closed essential surfaces or none at all, despite having arbitrarily large hyperbolic volume, resolving a question about the correlation between essential surfaces and volume in these lens space surgery candidates.

ABSTRACT

We construct an algorithm that lists all closed essential surfaces in the complement of a knot that lies on the fiber of a trefoil or figure eight knot. Such knots are Berge knots and hence admit lens space surgeries. Furthermore they may have arbitrarily large hyperbolic volume. Using this algorithm we concoct large volume Berge knots of two flavors: those whose complement contains arbitrarily many distinct closed essential surfaces, and those whose complement contains no closed essential surfaces.

Motivation & Objective

• To determine whether the presence of closed essential surfaces in Berge knot complements correlates with their hyperbolic volume.
• To classify all closed essential surfaces in the complements of knots lying on the fiber of a trefoil or figure-eight knot.
• To construct examples of Berge knots with arbitrarily large volume that either contain many distinct closed essential surfaces or none at all.
• To establish a computational framework based on continued fractions and surface invariants to detect such surfaces systematically.

Proposed method

• Adapts algorithms from Culler, Jaco, and Rubinstein and Floyd and Hatcher to list properly embedded incompressible surfaces in once-punctured torus bundles that are disjoint from a given level curve.
• Introduces Algorithm 6.1 to list all twisted surfaces in a once-punctured torus bundle that are essential in the complement of a level curve.
• Extends Algorithm 6.1 to Algorithm 6.2 to identify only those twisted surfaces whose boundaries are meridional curves of the knot binding the open book.
• Uses framing theory to ensure that capped-off surfaces from meridional-boundary twisted surfaces remain incompressible in the full knot complement.
• Applies a streamlined version of the algorithm (Algorithm 9.1) to Berge knots with continued fraction slopes satisfying alternating sign and magnitude constraints.
• Derives a key equation involving subset sums of continued fraction coefficients to determine the existence of closed essential surfaces.

Experimental results

Research questions

• RQ1 Can Berge knots with arbitrarily large hyperbolic volume contain arbitrarily many distinct closed essential surfaces in their complements?
• RQ2 Is there a structural correlation between the presence of closed essential surfaces and the hyperbolic volume of Berge knots?
• RQ3 Can Berge knots with arbitrarily large volume contain no closed essential surfaces at all?
• RQ4 What algebraic conditions on the continued fraction expansion of a knot's slope determine the existence or non-existence of closed essential surfaces in its complement?
• RQ5 How can one algorithmically classify all closed essential surfaces in the complements of knots on the fiber of a trefoil or figure-eight knot?

Key findings

• There exist Berge knots of arbitrarily large hyperbolic volume that contain arbitrarily many distinct closed essential surfaces in their complements, as demonstrated by constructing a family of knots with slope [z, -z, z, ..., ±z] of increasing length.
• For each such family, the number of distinct closed essential surfaces grows with the length of the continued fraction, and surfaces of different genera are obtained for different subsets of indices.
• There exist Berge knots of arbitrarily large volume with no closed essential surfaces in their complements, proven via a modular obstruction argument on the continued fraction coefficients.
• Knots with slope [Φₙ+2, -Φₙ, +Φₙ, ..., ±Φₙ] for large Φₙ are small (no closed essential surfaces) and have volumes tending to infinity as n increases.
• The existence of closed essential surfaces is determined by a specific equation involving subset sums of the continued fraction coefficients, with constraints on non-consecutive indices and inclusion of 1.
• The algorithm successfully detects all such surfaces by combining twisted surface theory with meridional boundary detection in once-punctured torus bundles.