[Paper Review] Bridge Number and the Curve Complex

Jesse Johnson|ArXiv.org|Mar 3, 2006

Geometric and Algebraic Topology11 references13 citations

TL;DR

This paper establishes that hyperbolic tunnel-number one knots with arbitrarily high bridge number exist by linking bridge number to distance in the curve complex of a genus-two surface. Using a complexity measure $ d(\tau) $ for unknotting tunnels, the authors prove that knots with $ d(\tau) > 5 $ have toroidal bridge number greater than one, and that such tunnels exist for arbitrarily large $ d(\tau) $, implying the existence of hyperbolic knots with high bridge number and non-one-bridge presentations over unknotted tori.

ABSTRACT

We show that there are hyperbolic tunnel-number one knots with arbitrarily high bridge number and that "most" tunnel-number one knots are not one-bridge with respect to an unknotted torus. The proof relies on a connection between bridge number and a certain distance in the curve complex of a genus-two surface.

Motivation & Objective

• To demonstrate that there exist hyperbolic, tunnel-number one knots with arbitrarily high bridge number $ b_0(K) $.
• To show that 'most' tunnel-number one knots are not one-bridge with respect to an unknotted torus, i.e., $ b_1(K) > 1 $.
• To establish a connection between the bridge number of a knot and the distance in the curve complex of a genus-two surface associated with its Heegaard splitting.
• To prove that knots with high curve complex distance $ d(\tau) > 5 $ have $ b_1(K) > 1 $, using the geometry of handlebody sets.
• To construct explicit examples of tunnel-number one knots with $ d(\tau) > N $ for any $ N $, ensuring hyperbolicity and high bridge number.

Proposed method

• Define a complexity invariant $ d(\tau) $ as the distance in the curve complex $ C(\Sigma) $ between the boundary of the meridian disk dual to the unknotting tunnel $ \tau $ and the handlebody set $ \mathbf{H_2} $ associated with the knot complement.
• Use the invariance of $ d(\tau) $ under tunnel isotopies to ensure it is a well-defined topological invariant of the knot and tunnel.
• Apply results from curve complex geometry, including $ \delta $-hyperbolicity and quasi-convexity, to bound distances and derive contradictions when assumptions on distance fail.
• Leverage Lemma 22 to construct a non-separating disk $ D_1 \subset H_1 $ such that $ d(\partial D_1, \mathbf{H_2}) > N+1 $, ensuring $ d(\tau) > N $.
• Use Thompson’s result that toroidal manifolds have Heegaard distance $ \leq 2 $ to show that if $ d(\tau) > 3 $, the knot complement is atoroidal, implying hyperbolicity.
• Combine the results using Morimoto’s theorem on connect sums to show that if $ d(\tau_1), d(\tau_2) > 5 $, then the connect sum has tunnel number 3.

Experimental results

Research questions

• RQ1 Can tunnel-number one knots have arbitrarily high bridge number with respect to a 2-sphere?
• RQ2 Are there hyperbolic tunnel-number one knots that are not one-bridge with respect to an unknotted torus?
• RQ3 How does the distance in the curve complex of a genus-two surface relate to the bridge number of a knot?
• RQ4 Can the curve complex distance $ d(\tau) $ be used to certify that $ b_1(K) > 1 $?
• RQ5 What is the relationship between high curve complex distance and the hyperbolicity of tunnel-number one knots?

Key findings

• For every integer $ N $, there exists a hyperbolic, tunnel-number one knot $ K $ such that $ b_0(K) > N $.
• If $ d(\tau) > 5 $, then $ b_1(K) > 1 $, meaning the knot is not one-bridge with respect to an unknotted torus.
• The curve complex distance $ d(\tau) $ provides a lower bound on the bridge number: $ b_0(K) \geq d(\tau) $.
• Knots with $ d(\tau) > 3 $ have atoroidal complements, so they are hyperbolic if not torus knots.
• The construction of such knots is generic: 'most' unknotting tunnels have high $ d(\tau) $, implying 'most' tunnel-number one knots are not one-bridge over unknotted tori.
• The connect sum of two tunnel-number one knots with $ d(\tau) > 5 $ has tunnel number 3, confirming non-minimality of tunnel systems under sum.