Amphichiral knot
In the mathematical field of knot theory , an amphichiral knot (also: achiral knot ) is a knot that is equivalent to its mirror image. A chiral knot is a knot that is not equivalent to its mirror image.
Amphichiral knots are important in biology and chemistry because the chirality or achirality of molecules is of great importance for their physical and chemical properties. Chirally knotted molecules in pharmaceutical products led to severe damage to newborns in the 1950s because mirror-inverted molecules often work in completely different ways. This problem does not occur with amphichirally knotted molecules.
Formal definition
A node is an equivalence class of embedding (or when), two embeddings are considered equivalent an orientation-preserving homeomorphism of (or are), which reflects the embedding into the other.
The mirror image of a node is obtained by applying an orientation-reversing homeomorphism to the image of an embedding representing the node. Because all orientation-reversing homeomorphisms of the (or ) are homotopic to one another, this definition does not depend on the choice of the orientation-reversing homeomorphism.
The mirror image of the node is denoted by.
A knot is called amphichiral (or achiral) if it is equivalent to its mirror image, i.e. if there is not only an orientation-reversing, but also an orientation-preserving homeomorphism, which maps the knot onto its mirror image.
A knot is said to be chiral if it is not equivalent to its mirror image.
Examples of chiral and amphichiral knots
- The trefoil loop is a chiral knot: the left-handed trefoil loop is not equivalent to the right-handed trefoil loop.
- More generally, all torus knots are chiral knots.
- The figure eight knot is amphichiral.
Invariants
The following invariants can distinguish a node from its mirror image:
- Signature : the signature of the mirror image has the opposite sign, the signature of an amphichiral knot is therefore zero. With this invariant one can distinguish the mirror images of torus knots (e.g. the clover leaf loop) from the original knots.
- Chern-Simons invariant : the Chern-Simons invariant of the mirror image has the opposite sign, the Chern-Simons invariant of an amphichiral knot is therefore zero.
- Number of crossings : the modulo 2 reduction of the number of crossings of a node is opposite to that of the mirror image, so the number of crossings of an amphichiral node is an even number
- Jones polynomial : the Jones polynomial of the mirror image is obtained by substituting into the Jones polynomial of , i. H. . The Jones polynomial of an amphichiral knot is therefore palindromic.
- τ-invariant from Ozsváth and Szabó, see Heegaard-Floer homology .
Related terms
Intersection number | 3 | 4th | 5 | 6th | 7th | 8th | 9 | 10 | 11 | 12 | 13 | 14th | 15th | 16 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Chiral knots | 1 | 0 | 2 | 2 | 7th | 16 | 49 | 152 | 552 | 2118 | 9988 | 46698 | 253292 | 1387166 |
Reversible knot | 1 | 0 | 2 | 2 | 7th | 16 | 47 | 125 | 365 | 1015 | 3069 | 8813 | 26712 | 78717 |
Completely chiral knots | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 27 | 187 | 1103 | 6919 | 37885 | 226580 | 1308449 |
Amphichiral knots | 0 | 1 | 0 | 1 | 0 | 5 | 0 | 13 | 0 | 58 | 0 | 274 | 1 | 1539 |
Positive amphichiral knots | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 6th | 0 | 65 |
Negative amphichiral knots | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 6th | 0 | 40 | 0 | 227 | 1 | 1361 |
Completely amphichiral knots | 0 | 1 | 0 | 1 | 0 | 4th | 0 | 7th | 0 | 17th | 0 | 41 | 0 | 113 |
literature
- Gilbert, ND; Porter, T .: Knots and surfaces. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, 1994. ISBN 0-19-853397-7 .
- Murasugi, Kunio: Knot theory & its applications. Translated from the 1993 Japanese original by Bohdan Kurpita. Reprint of the 1996 translation. Modern Birkhäuser Classics. Birkhauser Boston, Inc., Boston, MA, 2008. ISBN 978-0-8176-4718-6 .
Web links
- Amphichiral Knot (MathWorld)
Individual evidence
- ^ Ross S. Forgan / Jean-Pierre Sauvage / J. Fraser Stoddart, "Chemical Topology: Complex Molecular Knots, Links, and Entanglements", in: Chemical Reviews 111 (2011), 5434-5464
- ↑ Stereochemistry: Determining Molecular Chirality