Tunnel number

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.