For every knot there is a collection of disjoint properly embedded arcs in the knot exterior such that the complement of a neighborhood of the arcs and knot forms a solid handlebody. The minimum number of such arcs that is required is called the tunnel number of the knot.
The computation of the tunnel numbers of prime knots of 10 or fewer crossings was completed in: Morimoto, Kanji, Sakuma, Makoto, Yokota, Yoshiyuki,
Identifying tunnel number one knots,
J. Math. Soc. Japan 48 (1996), no. 4, 667--688.