On the intersection of unknotting tunnels and the decomposing annulus in connected sums

Details On the intersection of unknotting tunnels and the decomposing annulus in connected sums On the intersection of unknotting tunnels and the decomposing annulus in connected sums

2002-11-26

arxiv.org/abs/math/0211407v1

Mathematics

Geometric Topology

17 pages, 4 figures

Scientific paper

Given $(V_1,V_2)$ a Heegaard splitting of the complement of a composite knot $K=K_1# K_2$ in $S^3$, where $K_i, i=1,2$ are prime knots, we have a unique, up to isotopy, decomposing annulus $A$. When the intersection of $A$ and $V_1$ is a minimal collection of disks we study the components of $V_1-N(A)$ and show that at most one component is a 3-ball meeting $A$ in two disks. This is a crucial step in proving the conjecture that a necessary and sufficient condition for the tunnel number of a connected sum to be less than or equal to the sum of the tunnel numbers is that one of the knots has a Heegaard splitting in which a merdian curve is primitive.

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