In a mathematical field of knot theory

Knot theory

In topology, knot theory is the study of mathematical knots. While inspired by knots which appear in daily life in shoelaces and rope, a mathematician's knot differs in that the ends are joined together so that it cannot be undone. In precise mathematical language, a knot is an embedding of a...

, the bridge number is an invariant

Knot invariant

In the mathematical field of knot theory, a knot invariant is a quantity 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 invariants are indeed numbers, but invariants can range from the...

 of a knot. It is defined as the minimal number of bridges required in all the possible bridge representations of a knot. In bridge representation, a knot lies entirely in the plane apart for a finite number of bridges whose projections onto the plane are straight lines.

Equivalently the bridge number is the minimal number of local maxima of the projection of the knot onto a vector, where we minimize over all projections and over all conformations of the knot.

It can be shown that every n-bridge knot can be decomposed into two trivial n-tangles and hence 2-bridge knot

2-bridge knot

In the mathematical field of knot theory, a 2-bridge knot is a knot which can be isotoped so that the natural height function given by the z-coordinate has only two maxima and two minima as critical points....

s are rational knots.

Other numerical invariants

• Crossing number
  Crossing number (knot theory)
  In the mathematical area of knot theory, the crossing number of a knot is the minimal number of crossings of any diagram of the knot. It is a knot invariant....
  
  
• Linking coefficient
• Unknotting number
  Unknotting number
  In the mathematical area of knot theory, the unknotting number of a knot is the minimum number of times the knot must be passed through itself to untie it. If a knot has unknotting number n, then there exists a diagram of the knot which can be changed to unknot by switching n crossings...
  
  
• Stick number
  Stick number
  In the mathematical theory of knots, the stick number is a knot invariant that intuitively gives the smallest number of straight "sticks" stuck end to end needed to form a knot...