The tunnel number of the connect sum of $n$ knots is at least $n$
Martin Scharlemann and Jennifer Schultens
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.