In mathematics, the tunnel number of a

knot A knot is an intentional complication in Rope, cordage which may be practical or decorative, or both. Practical knots are classified by function, including List of hitch knots, hitches, List of bend knots, bends, List of loop knots, loop knots, ...

, as first defined by Bradd Clark, is a

knot invariant In the mathematical field of knot theory, a knot invariant is a quantity (in a broad sense) defined for each knot which is the same for equivalent knots. The equivalence is often given by ambient isotopy but can be given by homeomorphism. Some ...

, given by the minimal number of arcs (called tunnels) that must be added to the knot so that the complement becomes a

handlebody In the mathematical field of geometric topology, a handlebody is a decomposition of a manifold into standard pieces. Handlebodies play an important role in Morse theory, cobordism theory and the surgery theory of high-dimensional manifolds. Handle ...

. 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 In the mathematical field of geometric topology, a Heegaard splitting () is a decomposition of a compact oriented 3-manifold that results from dividing it into two handlebodies. Definitions Let ''V'' and ''W'' be handlebodies of genus ''g'', and ...

of the link exterior.

Examples

* The

unknot In the mathematical theory of knots, the unknot, not knot, or trivial knot, is the least knotted of all knots. Intuitively, the unknot is a closed loop of rope without a knot tied into it, unknotted. To a knot theorist, an unknot is any embe ...

is the only knot with tunnel number 0. * The

trefoil knot In knot theory, a branch of mathematics, the trefoil knot is the simplest example of a nontrivial knot. The trefoil can be obtained by joining together the two loose ends of a common overhand knot, resulting in a knotted loop. As the simplest kn ...

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

* *. *. *. *. Knot invariants {{topology-stub