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From Wikipedia, the free encyclopedia

 A trefoil knot, drawn with bridge number 2
A trefoil knot, drawn with bridge number 2

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.

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Transcription

You'd have a hard time finding Königsberg on any modern maps, but one particular quirk in its geography has made it one of the most famous cities in mathematics. The medieval German city lay on both sides of the Pregel River. At the center were two large islands. The two islands were connected to each other and to the river banks by seven bridges. Carl Gottlieb Ehler, a mathematician who later became the mayor of a nearby town, grew obsessed with these islands and bridges. He kept coming back to a single question: Which route would allow someone to cross all seven bridges without crossing any of them more than once? Think about it for a moment. 7 6 5 4 3 2 1 Give up? You should. It's not possible. But attempting to explain why led famous mathematician Leonhard Euler to invent a new field of mathematics. Carl wrote to Euler for help with the problem. Euler first dismissed the question as having nothing to do with math. But the more he wrestled with it, the more it seemed there might be something there after all. The answer he came up with had to do with a type of geometry that did not quite exist yet, what he called the Geometry of Position, now known as Graph Theory. Euler's first insight was that the route taken between entering an island or a riverbank and leaving it didn't actually matter. Thus, the map could be simplified with each of the four landmasses represented as a single point, what we now call a node, with lines, or edges, between them to represent the bridges. And this simplified graph allows us to easily count the degrees of each node. That's the number of bridges each land mass touches. Why do the degrees matter? Well, according to the rules of the challenge, once travelers arrive onto a landmass by one bridge, they would have to leave it via a different bridge. In other words, the bridges leading to and from each node on any route must occur in distinct pairs, meaning that the number of bridges touching each landmass visited must be even. The only possible exceptions would be the locations of the beginning and end of the walk. Looking at the graph, it becomes apparent that all four nodes have an odd degree. So no matter which path is chosen, at some point, a bridge will have to be crossed twice. Euler used this proof to formulate a general theory that applies to all graphs with two or more nodes. A Eulerian path that visits each edge only once is only possible in one of two scenarios. The first is when there are exactly two nodes of odd degree, meaning all the rest are even. There, the starting point is one of the odd nodes, and the end point is the other. The second is when all the nodes are of even degree. Then, the Eulerian path will start and stop in the same location, which also makes it something called a Eulerian circuit. So how might you create a Eulerian path in Königsberg? It's simple. Just remove any one bridge. And it turns out, history created a Eulerian path of its own. During World War II, the Soviet Air Force destroyed two of the city's bridges, making a Eulerian path easily possible. Though, to be fair, that probably wasn't their intention. These bombings pretty much wiped Königsberg off the map, and it was later rebuilt as the Russian city of Kaliningrad. So while Königsberg and her seven bridges may not be around anymore, they will be remembered throughout history by the seemingly trivial riddle which led to the emergence of a whole new field of mathematics.

Contents

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 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.

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.[3]

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, 151, American Mathematical Society, Providence, RI, p. 129, ISBN 978-1-4704-1020-9, MR 3203728 .
  3. ^ Schultens, Jennifer (2003), "Additivity of bridge numbers of knots", Mathematical Proceedings of the Cambridge Philosophical Society, 135 (3): 539–544, doi:10.1017/S0305004103006832, MR 2018265 .

Further reading

This page was last edited on 20 June 2017, at 01:14.
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