Tunnel number degeneration under the connected sum of prime knots

Abstract

We study 2-string free tangle decompositions of knots with tunnel number two. As an application, we construct infinitely many counter-examples to a conjecture in the literature stating that the tunnel number of the connected sum of prime knots doesn't degenerate by more than one.

40 pages, 28 figures; Version 2: reference added, minor changes in text

References in corpus (5)

• Heegaard splittings of knot exteriors
• Incompressible surfaces and (1,2)-knots
• On the degeneration ratio of tunnel numbers and free tangle decompositions of knots
• Hyperbolic (1,2)-knots in S^3 with crosscap number two and tunnel number one
• On Heegaard splittings of glued 3-manifolds

Cited by in corpus (4)

• A lower bound on tunnel number degeneration
• Prime knot complements with meridional essential surfaces of arbitrarily high genus
• Equivariant Heegaard genus of reducible 3-manifolds
• On the degeneration of tunnel numbers under connected sum