In knot theory, a chiral knot is a knot that is not equivalent to its mirror image. A oriented node equivalent to its specular reflection is called an amphichiral node or achiral node . The chirality of a node is an invariant of a node . The chirality of the nodes can be further classified depending on whether it is reversible or not.
There are only 5 types of node symmetries determined by chirality and reversibility - completely chiral, reversible, positively amphichiral, irreversible, negatively amphichiral, irreversible and completely amphichiral reversible [1] .
Content
Background
The chirality of some nodes has long been suspected and proved by Max Den in 1914. P. G. Tet conjectured that all amphichiral nodes have an even number of intersections , but found a counterexample in 1998 [2] . However, the Tate conjecture was proved for simple alternate nodes [3] .
| Number of intersections | 3 | four | five | 6 | 7 | eight | 9 | ten | eleven | 12 | 13 | 14 | 15 | sixteen | OEIS sequence |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Chiral nodes | one | 0 | 2 | 2 | 7 | sixteen | 49 | 152 | 552 | 2118 | 9988 | 46698 | 253292 | 1387166 | N / a |
| Bilateral nodes | one | 0 | 2 | 2 | 7 | sixteen | 47 | 125 | 365 | 1015 | 3069 | 8813 | 26712 | 78717 | A051769 |
| Fully chiral nodes | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 27 | 187 | 1103 | 6919 | 37885 | 226580 | 1308449 | A051766 |
| Amphichiral nodes | 0 | one | 0 | one | 0 | five | 0 | 13 | 0 | 58 | 0 | 274 | one | 1539 | A052401 |
| Positive amphihiral nodes | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | one | 0 | 6 | 0 | 65 | A051767 |
| Negatively amphichiral nodes | 0 | 0 | 0 | 0 | 0 | one | 0 | 6 | 0 | 40 | 0 | 227 | one | 1361 | A051768 |
| Completely amphichiral nodes | 0 | one | 0 | one | 0 | four | 0 | 7 | 0 | 17 | 0 | 41 | 0 | 113 | A052400 |
- Both possible shamrocks .
Left Shamrock.
Right shamrock.
The simplest chiral knot is the trefoil , the chirality of which was shown by Max Dan . All toric knots are chiral. The Alexander polynomial cannot determine the chirality of the knot, but the Jones polynomial in some cases can. If V k ( q ) ≠ V k ( q −1 ), then the node is chiral, but the converse is not necessarily true. even better recognizes chirality, but so far there is no known polynomial invariant of a node that would completely determine chirality [4] .
Two-way node
A reversible chiral node is called bilateral [5] . Among the examples of bilateral nodes is the trefoil.
Fully Chiral Node
If a node is not equivalent to either its inverse or its mirror image, it is called completely chiral, an example is node 9 32 [5] .
Amphichiral node
An amphichiral node is a node having an auto-homeomorphism of an α 3-sphere that reverses orientation and fixes the node as a set.
All amphichiral alternating ones have an even number of intersections . The first amphichiral node with an odd number of intersections, namely with 15 intersections, was found by Hoste and others. [3]
Total Amphichirality
If a node is isotopic to its inverse and its mirror image, it is called completely amphichiral. The simplest node with this property is the eight .
Positive amphihirality
If the auto-homeomorphism α preserves the orientation of the knot, they speak of positive amphichirality. This is equivalent to the isotopy of the site to its mirror image. None of the nodes with the number of intersections less than twelve is positively amphichiral [5] .
Negative Amphichirality
If the auto-homeomorphism α reverses the orientation of the knot, they speak of negative amphichirality. This is equivalent to the isotopicity of the site inverse mirror reflection. A node with this property with a minimum number of intersections is 8 17 [5] .
Notes
- ↑ Hoste, Thistlethwaite, Weeks, 1998 , p. 33-48.
- ↑ Jablan, Slavik & Sazdanovic, Radmila. “ History of Knot Theory and Certain Applications of Knots and Links ”, LinKnot .
- ↑ 1 2 Weisstein, Eric W. Amphichiral Knot ( Wolfram ) on Wolfram MathWorld . Accessed: May 5, 2013.
- ↑ “Chirality of Knots 9 42 and 10 71 and Chern-Simons Theory” by P. Ramadevi, TR Govindarajan, and RK Kaul
- ↑ 1 2 3 4 Three Dimensional Invariants Knot Atlas
Literature
- Jim Hoste, Morwen Thistlethwaite, Jeff Weeks. The first 1,701,936 knots // The Mathematical Intelligencer. - 1998. - T. 20 , no. 4 . - DOI : 10.1007 / BF03025227 . Archived December 15, 2013.