In mathematics, the tunnel number of a knot, as first defined by Bradd Clark, is a knot invariant, given by the minimal number of arcs (called tunnels) that must be added to the knot so that the complement becomes a handlebody. The tunnel number can equally be defined for links. The boundary of a regular neighbourhood of the union of the link and its tunnels forms a Heegaard splitting of the link exterior.
Examples The unknot is the only knot with tunnel number 0. The trefoil knot has tunnel number 1. In general, any nontrivial torus knot has tunnel number 1. Every link L has a tunnel number. This can be seen, for example, by adding a 'vertical' tunnel at every crossing in a diagram of L. It follows from this construction that the tunnel number of a knot is always less than or equal to its crossing number.
References
Clark, Bradd (1980), "The Heegaard Genus Of Manifolds Obtained By Surgery On Links And Knots", International Journal of Mathematics and Mathematical Sciences, 3 (3): 583–589, doi:10.1155/S0161171280000440 Boileau, Michel; Lustig, Martin; Moriah, Yoav (1994), "Links with super-additive tunnel number", Mathematical Proceedings of the Cambridge Philosophical Society, 115 (1): 85–95, Bibcode:1994MPCPS.115...85B, doi:10.1017/S0305004100071930, MR 1253284. Kobayashi, Tsuyoshi; Rieck, Yo'av (2006), "On the growth rate of the tunnel number of knots", Journal für die reine und angewandte Mathematik, 2006 (592): 63–78, arXiv:math/0402025, doi:10.1515/CRELLE.2006.023, MR 2222730. Scharlemann, Martin (1984), "Tunnel number one knots satisfy the Poenaru conjecture", Topology and Its Applications, 18 (2–3): 235–258, doi:10.1016/0166-8641(84)90013-0, MR 0769294. Scharlemann, Martin (2004), "There are no unexpected tunnel number one knots of genus one", Transactions of the American Mathematical Society, 356 (4): 1385–1442, arXiv:math/0106017, doi:10.1090/S0002-9947-03-03182-9, MR 2034312.