| 词条 | Bridge number |
| 释义 |
In the mathematical field of knot theory, the bridge number is an invariant of a knot defined as the minimal number of bridges required in all the possible bridge representations of a knot. DefinitionGiven a knot or link, draw a diagram of the link using the convention that a gap in the line denotes an undercrossing. Call an 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 number was first studied in the 1950s by Horst Schubert.[2] 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. PropertiesEvery 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.[3] Other numerical invariants
References1. ^1 {{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}}. 2. ^{{citation | last = Schultens | first = Jennifer | isbn = 978-1-4704-1020-9 | mr = 3203728 | page = 129 | publisher = American Mathematical Society, Providence, RI | series = Graduate Studies in Mathematics | title = Introduction to 3-manifolds | url = https://books.google.com/books?id=Qt7DAwAAQBAJ&pg=PA129 | volume = 151 | year = 2014}}. 3. ^{{citation | last = Schultens | first = Jennifer | 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 }}. Further reading
1 : Knot invariants |
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