Additive tunnel number and primitive elements

1999/06/10 by Yoav Moriah, Moriah, Yoav
Mathematics · #FOS: Mathematics #Geometric Topology (math.GT) #math.GT

paper · pdf · doi:10.48550/arxiv.math/9906061

20 pages, 11 figures

arxiv created 1999/06/10 · arxiv updated 2009/11/30

Abstract

It is proven here that if the connected sum of two tunnel number one knots in the 3-sphere is a tunnel number two knot, then at least one of the summand knots has a genus two Heegaard splitting with a meridian as a primitive element. Hence this is a necessary and sufficient condition for tunnel number one knots to have additive tunnel number.

Related

• The tunnel number of the sum of n knots is at least n
• On the degeneration of tunnel numbers under connected sum
• Iterated splitting and the classification of knot tunnels
• On Heegaard splittings of knot exteriors with tunnel number degenerations
• Middle tunnels by splitting