Bridge number

This article is about a mathematical concept. For the telecommunications term, see Conference Call.
A trefoil knot, drawn with bridge number 2

In the mathematical field of knot theory, the bridge number, also called the bridge index, is an invariant of a knot defined as the minimal number of bridges required in all the possible bridge representations of a knot.

Definition

Given a knot or link, draw a diagram of the link using the convention that a gap in the line denotes an undercrossing. Call an unbroken arc in this diagram a bridge if it includes at least one overcrossing. Then the bridge number of a knot can be found as the minimum number of bridges required for any diagram of the knot.[1] Bridge numbers were first studied in the 1950s by Horst Schubert.[2] [3]

The bridge number can equivalently be defined geometrically instead of topologically. 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. In this context, the bridge number is often called the crookedness.

Properties

Every non-trivial knot has bridge number at least two,[1] so the knots that minimize the bridge number (other than the unknot) are the 2-bridge knots. It can be shown that every n-bridge knot can be decomposed into two trivial n-tangles and hence 2-bridge knots are rational knots.

If K is the connected sum of K1 and K2, then the bridge number of K is one less than the sum of the bridge numbers of K1 and K2.[4]

Other numerical invariants

References

  1. ^ a b Adams, Colin C. (1994), The Knot Book, American Mathematical Society, p. 65, ISBN 9780821886137.
  2. ^ Schultens, Jennifer (2014), Introduction to 3-manifolds, Graduate Studies in Mathematics, vol. 151, American Mathematical Society, Providence, RI, p. 129, ISBN 978-1-4704-1020-9, MR 3203728.
  3. ^ Schubert, Horst (December 1954). "Über eine numerische Knoteninvariante". Mathematische Zeitschrift. 61 (1): 245–288. doi:10.1007/BF01181346.
  4. ^ Schultens, Jennifer (2003), "Additivity of bridge numbers of knots", Mathematical Proceedings of the Cambridge Philosophical Society, 135 (3): 539–544, arXiv:math/0111032, Bibcode:2003MPCPS.135..539S, doi:10.1017/S0305004103006832, MR 2018265.

Further reading