Alternating node diagram

An alternating knot 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.

Flype: a tangle within the node diagram is rotated by 180 ⁰

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 . ${\ displaystyle g}$ ${\ displaystyle d = 2g}$ ${\ displaystyle d}$

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: ${\ displaystyle tw (D)}$

{\ displaystyle {\ frac {1} {2}} v_ {3} tw (D) -1 <vol (S ^ {3} \ setminus K) <10v_ {3} (tw (D) -1)}

,

where denotes the Gieseking constant . ${\ displaystyle v_ {3}}$

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.

[1] Kunio Murasugi: Jones polynomials and classical conjectures in knot theory. Topology, 26 (1987) no. 2, 187-194.

[2] 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.

[3] Kunio Murasugi : On the genus of the alternating knot. I, II. J. Math. Soc. Japan 10 1958 94-105, 235-248.

[4] Richard Crowell : Genus of alternating link types. Ann. of Math. (2) 69 1959 258-275.

[5] David Gabai : Genera of the alternating links. Duke Math. J. 53 (1986) no. 3, 677-681.

[6] William Menasco: Closed incompressible surfaces in alternating knot and link complements. Topology, 23, no. 1, 37-44 (1984).

[7] 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.