The tunnel number of the sum of \(n\) knots is at least \(n\). (English) Zbl 0929.57003

Scharlemann, Martin; Schultens, Jennifer

Topology 38, No. 2, 265-270 (1999).

The tunnel number of a knot \(K\) in \(S^3\) is the minimum number of disjoint proper tunnels needed to be drilled into the complement of a tubular neighborhood of \(K\) in order to obtain a handlebody. It has been known for some time now that if a knot has tunnel number \(1\) then it is prime. This result is here generalized to show that if the tunnel number of \(K\) is \(n\) then \(K\) can be expressed as a connected sum of \(\leq n\) prime knots.

Reviewer: J.Levine (Waltham)

MSC:

57M25

Knots and links in the \(3\)-sphere (MSC2010)

Keywords:

Heegaard decomposition