bridge number (Rev #4, changes) in nLab

Showing changes from revision #3 to #4: Added | Removed | Changed

Bridge number

Definitions

A bridge in a knot diagram is an arc that is the overpass in at least one crossing.

The bridge number, b(K)b(K) of a knot KK is the minimum number of bridges occuring in a diagram of the knot.

By convention the unknot has bridge number equal to 11.

Why not 00? (I don't see any bridges in the obvious diagram.) Are there numerical operations on bridge numbers that only work for the unknot if its bridge number is 11? Can we motivate the existence of a bridge in the unknot by proper application of negative thinking?

Example

The usual picture of the trefoil knot, which we will write as T 2,3T_{2,3} , has three bridges, but the trefoil knot actually has bridge number 2, as shown: (two trefoils).

has The three first bridges, drawing but corresponds to the trefoil picture knot as actually a has (2,3)- bridge number 2.torus knot. The knot thus can be represented on the surface of a solid torus S 1×D 2S^1\times D^2 in S 3S^3. The 3-sphere can be represented as the union of two solid tori with, in this case, the second one being D 2×S 1D^2\times S^1, where the boundary of the D 2D^2 in each case is the corresponding S 1S^1 of the other case. Thinking of the trefoil on this other solid torus it is a (3,2)(3,2)-torus knot …. (the second picture) and this has just two bridges, so b(T 2,3)≤2b(T_{2,3})\leq 2.

The How first does drawing one corresponds know to that the picture trefoil as does a not (2,3)- havetorus knotb(K)=1b(K) = 1 . The For knot that thus we can use be represented on the surface of a solid torusS 1×D 2S^1\times D^2 in S 3S^3. The 3-sphere can be represented as the union of two solid tori with, in this case, the second one being D 2×S 1D^2\times S^1, where the boundary of the D 2D^2 in each case is the corresponding S 1S^1 of the other case. Thinking of the trefoil on this other solid torus it is a (3,2)(3,2)-torus knot …. (the second picture) and this has just two bridges, so b(T 2,3)≤2b(T_{2,3})\leq 2.

How does one know that the trefoil does not have b(K)=1b(K) = 1. For that we use

To complete the proof one needs to show or know that the trefoil is a non-trivial knot. That is most simply done using some invariant such as 3-colorability?.

(The picture on this page is from theMaths and Knots Exhibition website, of the Centre for the Popularisation of Mathematics. Their assistence is acknowledged.)