Tunnel Number

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 required number of such arcs is the tunnel number of the knot.

The computation of the tunnel numbers of prime knots of 10 or fewer crossings was completed in [1].

For 11 and 12 crossing alternating knots, along with many nonalternating knots, results were provided by [3].

Specific Knots

12n242, otherwise known as P(-2,3,7)
   It was found in [2] that this knot has exactly 4 unknotting tunnels, so its tunnel number is 1.

References

[1] Morimoto, K. et. al., Identifying tunnel number one knots, J. Math. Soc. Japan 48 no. 4 (1996), 667-688.

[2] Heath, D. and Song, H-J., "Unknotting Tunnels for P(-2,3,7)," Journal of Knot Theory and its Ramifications, Vol 14, No. 8 (2005), 1077-1085.

[3] Castellano-Macías, F. and Owad, N., "The tunnel number of all 11 and 12 crossing alternating knots," arXiv preprint.