Bridge number
{{otheruses4|a mathematical concept|the telecommunications term|Conference Call}}
File:2-bridge trefoil.svg, 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.{{citation|title=The Knot Book|first=Colin C.|last=Adams|authorlink=Colin Adams (mathematician)|publisher=American Mathematical Society|year=1994|isbn=9780821886137|page=65|url=https://books.google.com/books?id=M-B8XedeL9sC&pg=PA65}}. Bridge numbers were first studied in the 1950s by Horst Schubert.{{citation
| last = Schultens | first = Jennifer
| isbn = 978-1-4704-1020-9
| mr = 3203728
| publisher = American Mathematical Society, Providence, RI
| series = Graduate Studies in Mathematics
| title = Introduction to 3-manifolds
| title-link = Introduction to 3-Manifolds
| at = [https://books.google.com/books?id=Qt7DAwAAQBAJ&pg=PA129 p. 129]
| volume = 151
| year = 2014}}.
|last1=Schubert
|first1=Horst
|title=Über eine numerische Knoteninvariante
|journal=Mathematische Zeitschrift
|date=December 1954
|volume=61
|issue=1
|pages=245–288
|doi=10.1007/BF01181346}}
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, 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.{{citation
| last = Schultens | first = Jennifer | author-link = Jennifer Schultens
| doi = 10.1017/S0305004103006832
| issue = 3
| journal = Mathematical Proceedings of the Cambridge Philosophical Society
| mr = 2018265
| pages = 539–544
| title = Additivity of bridge numbers of knots
| volume = 135
| year = 2003| arxiv = math/0111032
| bibcode = 2003MPCPS.135..539S
}}.
Other numerical invariants
References
{{reflist}}
Further reading
- Cromwell, Peter (1994). Knots and Links. Cambridge. {{isbn|9780521548311}}.
{{Knot theory|state=collapsed}}