In knot theory, the number of intersections of a node is the smallest number of intersections on any diagram of a node. The number of intersections is a knot invariant .
Content
Examples
As an example: a trivial knot has zero intersections, the number of trefoil intersections is three, and the number of eight intersections is four. There are no more nodes with four or fewer intersections, and there are only two nodes with five intersections, but the number of nodes with specific intersection numbers increases rapidly as the number of intersections increases.
Tables
Tables of simple knots are traditionally indexed by the number of intersections with an additional description of which node from the set of nodes with a given number of intersections is meant (this ordering is not based on any properties, with the exception of toric nodes for which twisted nodes are listed first). The list starts with 3 1 (trefoil), 4 1 (eight), 5 1 , 5 2 , 6 1 , and so on. This order has not changed significantly since Theta , which published the table in 1877 [1] .
Additivity
There is very little progress in understanding the behavior of the number of intersections in elementary operations on nodes. The big open question is whether the number of intersections is additive with respect to the concatenation operation. It is also expected that the satellite node of node K will have more intersections than K , but this is not proven.
The additivity of the number of intersections of the concatenation of nodes has been proved for special cases, for example, if the source nodes are alternate [2] or if the source nodes are toric [3] [4] . Mark Lakenby gave a proof that there exists a constant N > 1 such that , but his method using cannot improve N to 1 [5] .
Application in bioinformatics
There is a strange connection between the number of intersections of a node and the physical behavior of DNA nodes. For simple DNA nodes, the number of intersections is a good predictor of the relative speed of the agarose gel electrophoresis DNA node. Basically, a higher number of intersections leads to a higher relative velocity [6] .
Related Invariants
There are related concepts of the and the asymptotic number of intersections. Both of these concepts define the boundaries of the standard number of intersections. There is a hypothesis that the asymptotic number of intersections is equal to the number of intersections.
Other numerical invariants of the node include the , the gearing coefficient , the number of segments, and the .
Notes
- ↑ Tait, 1898 , p. 273—347.
- ↑ Adams, 2004 , p. 69.
- ↑ Gruber, 2003 .
- ↑ Diao, 2004 , p. 857-866.
- ↑ Lackenby, 2009 , p. 747-768.
- ↑ Jonathan, 1996 , p. 39-58.
Literature
- Simon Jonathan. Energy functions for knots: Beginning to predict physical behavior // Mathematical Approaches to Biomolecular Structure and Dynamics / Jill P. Mesirov, Klaus Schulten, De Witt Sumners. - 1996. - T. 82. - (The IMA Volumes in Mathematics and its Applications). - DOI : 10.1007 / 978-1-4612-4066-2_4 .
- PG Tait. On Knots I, II, III // Scientific papers. - Cambridge University Press, 1898 .-- T. 1.
- CA Adams. The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots . - American Mathematical Society, 2004. - ISBN 9780821836781 .
- H. Gruber. Estimates for the minimal crossing number. - 2003 .-- arXiv : math / 0303273 .
- Yuanan Diao. The additivity of crossing numbers // Journal of Knot Theory and its Ramifications. - 2004. - T. 13 , no. 7 . - DOI : 10.1142 / S0218216504003524 .
- Marc Lackenby. The crossing number of composite knots // Journal of Topology. - 2009. - T. 2 , no. 4 . - DOI : 10.1112 / jtopol / jtp028 .