Notation of the doe car

Figure of projection of the knot numbered for the notation of the doe car

One of the methods to display a knot in the knot theory that is a topological one minute field with the notation (indication of the doe car, Dowker notation) of the doe car. The name comes from the Clifford Hugh doe car (English version) of the mathematician. Curl Friedrich gauss devised the idea to express a figure of projection of the knot which caused it by even progression, and improvement was added afterwards [¹].

The notations of the knot include a notation (English version) of Conway or the notation by a group of braided cord elsewhere.

Concrete notation

By the notation of the doe car, I express a figure of projection of the knot to have the point of intersection of the n unit toward by the even progression of the n unit.

Specifically, I let progression cope for a figure of projection of the knot to do it as follows, and to have the point of intersection of the n unit toward. At first attack the direction for a given knot, and one takes an initial point and the point where it is in the place that is not the point of intersection in the figure of projection of the knot. I trace the ingredient of the knot of the projection on the chart along the direction that I acquired some time ago from the point, but wave a number of 1 to the point of intersection to go along first then. Whenever just follow an ingredient along a direction, and go along the point of intersection; 2, 3, 4, … I number と order and continue it until I go around a knot and come back to the initial point. But I touch a mark of - to the number when I acquire an even number when an ingredient following goes along the ingredient of the upper part of the point of intersection.

While they go around a knot, about one point of intersection, by ingredient of the upper part and ingredient of the bottom and once go along the twice in total. Therefore, two numbers run out on one point of intersection, and an illustration of projection will have the number of the 2n unit (I refer to the right figure). 1, 3, 5 that pay attention to an odd number and an even number being dumped one by one by all means then on each point of intersection, and are an odd number, …I make the progression that set the other even number dumped on the point of intersection where a number of, 2n -1 is dumped sequentially. The progression in case of the right figure

{6,-12, 2, 8,-4,-10}

となる. In this way, I can let even progression support a figure of projection of the knot (but even the same knot may be displayed by different progression depending on how to get initial points).

I can restore an illustration of projection of the knot from even progression by following the above-mentioned operation adversely. Even if an illustration of projection of the different knot might be restored from the same progression then, but knots more than different three were not restored, and was able to restore the knot of two different again; they of the reflected image come to it-affiliated. Therefore the same knot is always provided from the same progression if it is a both hands type knot [²].

I can let the even progression that I pick quarrel and classified an end into for the eyes cope when I apply this method.

Footnote

1. For ^ Mitsuyuki Ochiai, Emiko Toyota, Shuji Yamada "introduction to knot theory Makino Bookstore with the computer", 1,996 years, it is page 54. ISBN 978-4,795,201,095。
2. A certain ^ knot and reflected image of the knot call the knot a both hands type knot at the time of equivalent. For example, a figure of eight knot is a both hands type knot, but the honewort knot is not so.

References

• For written by C, C Adams, Taizo Kanenobu reason "mathematics 培風館 of the knot", 1,998 years, it is 35-41 pages. ISBN 978-4,563,002,541。
• For Kunio Murasugi "application Japan criticism company with a knot theory", 1,993 years, it is 26-29 pages. ISBN 978-4,535,781,993。

It is acquired by "notation &oldid=46992915 of the https://ja.wikipedia.org/w/index.php?title= doe car"

This article is taken from the Japanese Wikipedia Notation of the doe car

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