THE TUNNEL NUMBER OF THE CONNECT SUM OF n KNOTS IS AT LEAST n

MARTIN SCHARLEMANN AND JENNIFER SCHULTENS

Abstract. Norwood showed that tunnel number 1 knots are prime. This led to the more general conjecture that the tunnel number of a connected sum of n knots is at least n. Here we prove this conjecture. The idea is to show that the splitting surface of a Heegaard splitting corresponding to a tunnel system realizing the tunnel number of the connected sum of n knots intersects each individual knot complement essentially. Then a sophisticated Euler characteristic argument, based on the idea of untelescoping the Heegaard splitting, yields the result.