Alternating node diagram
In the mathematical field of knot theory , an alternating knot diagram is a knot diagram that alternates between crossings and crossings . Analogously, an alternating interlocking diagram is an interlinking diagram for which crossovers and crossovers alternate when passing through each component.
An alternating node is a node that can be projected into an alternating node diagram in the plane. (Not every projection has to give an alternating diagram.) Similarly, a link is an alternating link if it has an alternating link diagram .
Intersection number
An alternating diagram is called reduced if there are four different regions at each intersection. A reduced alternating diagram calculates the number of intersections , i.e. H. it is the diagram of the minimum number of crossings for the given node.
Uniqueness
Two reduced alternating diagrams of the same oriented node emerge from a sequence of flypes (rotations of a tangle by 180 °) apart. This makes it easy to decide whether two alternating diagrams represent the same node.
Seifert area
The Seifert algorithm delivers a Seifert surface of minimal gender for alternating knots and entanglements . In particular, the equation applies to alternating links , where denotes the degree of the Alexander polynomial .
Twist number and hyperbolic volume
Alternating prime nodes are hyperbolic . The hyperbolic volume of a hyperbolic node can be estimated by the number of twist (i.e. number of twist regions) of an alternating graph:
- ,
where denotes the Gieseking constant .
literature
- Colin Adams: The Knot Book: An elementary introduction to the mathematical theory of knots. American Mathematical Society, Providence, RI, 2004, ISBN 0-8218-3678-1 .
Web links
- Alternating Knot (MathWorld)
Individual evidence
- ↑ Kunio Murasugi: Jones polynomials and classical conjectures in knot theory. Topology, 26 (1987) no. 2, 187-194.
- ↑ William Menasco, Morwen Thistlethwaite: The Tait FLYPING conjecture. Bull. Amer. Math. Soc. (NS) 25 (1991), no. 2, 403-412; The classification of alternating links. Ann. of Math. (2) 138 (1993) no. 1, 113-171.
- ↑ Kunio Murasugi : On the genus of the alternating knot. I, II. J. Math. Soc. Japan 10 1958 94-105, 235-248.
- ^ Richard Crowell : Genus of alternating link types. Ann. of Math. (2) 69 1959 258-275.
- ↑ David Gabai : Genera of the alternating links. Duke Math. J. 53 (1986) no. 3, 677-681.
- ^ William Menasco: Closed incompressible surfaces in alternating knot and link complements. Topology, 23, no. 1, 37-44 (1984).
- ↑ Marc Lackenby : The volume of hyperbolic alternating link complements. With an appendix by Ian Agol and Dylan Thurston. Proc. London Math. Soc. (3) 88 (2004), no. 1, 204-224.