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- Alternating_knot abstract "In knot theory, a knot or link diagram is alternating if the crossings alternate under, over, under, over, as one travels along each component of the link. A link is alternating if it has an alternating diagram.Many of the knots with crossing number less than 10 are alternating. This fact and useful properties of alternating knots, such as the Tait conjectures, was what enabled early knot tabulators, such as Tait, to construct tables with relatively few mistakes or omissions. The simplest non-alternating prime knots have 8 crossings (and there are three such: 819, 820, 821).It is conjectured that as the crossing number increases, the percentage of knots that are alternating goes to 0 exponentially quickly. Alternating links end up having an important role in knot theory and 3-manifold theory, due to their complements having useful and interesting geometric and topological properties. This led Ralph Fox to ask, "What is an alternating knot?" By this he was asking what non-diagrammatic properties of the knot complement would characterize alternating knots.Various geometric and topological information is revealed in an alternating diagram. Primeness and splittability of a link is easily seen from the diagram. The crossing number of a reduced, alternating diagram is the crossing number of the knot. This last is one of the celebrated Tait conjectures. An alternating knot diagram is in one to one correspondence with a planar graph. Each crossing is associated with an edge and half of the connected components of the complement of the diagram are associated with vertices in a checker board manner. File:Trefle.jpgFile:Frise.jpg".
- Alternating_knot thumbnail Knot_8sb19.svg?width=300.
- Alternating_knot wikiPageExternalLink www.entrelacs.net.
- Alternating_knot wikiPageID "1236143".
- Alternating_knot wikiPageLength "4563".
- Alternating_knot wikiPageOutDegree "32".
- Alternating_knot wikiPageRevisionID "681268186".
- Alternating_knot wikiPageWikiLink Category:Alternating_knots_and_links.
- Alternating_knot wikiPageWikiLink Category:Knot_invariants.
- Alternating_knot wikiPageWikiLink Crossing_number_(knot_theory).
- Alternating_knot wikiPageWikiLink Flype.
- Alternating_knot wikiPageWikiLink Haken_manifold.
- Alternating_knot wikiPageWikiLink Hyperbolic_geometry.
- Alternating_knot wikiPageWikiLink Hyperbolic_link.
- Alternating_knot wikiPageWikiLink Hyperbolization_theorem.
- Alternating_knot wikiPageWikiLink K._Murasugi.
- Alternating_knot wikiPageWikiLink Knot_(mathematics).
- Alternating_knot wikiPageWikiLink Knot_complement.
- Alternating_knot wikiPageWikiLink Knot_diagram.
- Alternating_knot wikiPageWikiLink Knot_theory.
- Alternating_knot wikiPageWikiLink Link_(knot_theory).
- Alternating_knot wikiPageWikiLink Louis_Kauffman.
- Alternating_knot wikiPageWikiLink Marc_Lackenby.
- Alternating_knot wikiPageWikiLink Morwen_Thistlethwaite.
- Alternating_knot wikiPageWikiLink Planar_graph.
- Alternating_knot wikiPageWikiLink Prime_knot.
- Alternating_knot wikiPageWikiLink Ralph_Fox.
- Alternating_knot wikiPageWikiLink Reduced_diagram.
- Alternating_knot wikiPageWikiLink Split_link.
- Alternating_knot wikiPageWikiLink Tait_conjectures.
- Alternating_knot wikiPageWikiLink Torus_knot.
- Alternating_knot wikiPageWikiLink Torus_link.
- Alternating_knot wikiPageWikiLink William_Menasco.
- Alternating_knot wikiPageWikiLink William_Thurston.
- Alternating_knot wikiPageWikiLink Writhe.
- Alternating_knot wikiPageWikiLink File:Frise.jpg.
- Alternating_knot wikiPageWikiLink File:Knot_8sb19.svg.
- Alternating_knot wikiPageWikiLink File:Trefle.jpg.
- Alternating_knot wikiPageWikiLinkText "Alternating knot".
- Alternating_knot wikiPageWikiLinkText "alternating diagrams".
- Alternating_knot wikiPageWikiLinkText "alternating knot".
- Alternating_knot wikiPageWikiLinkText "alternating".
- Alternating_knot wikiPageWikiLinkText "non-alternating knots".
- Alternating_knot hasPhotoCollection Alternating_knot.
- Alternating_knot id "AlternatingKnot".
- Alternating_knot id "TaitsKnotConjectures".
- Alternating_knot title "Alternating Knot".
- Alternating_knot title "Tait's Knot Conjectures".
- Alternating_knot wikiPageUsesTemplate Template:Citation_needed.
- Alternating_knot wikiPageUsesTemplate Template:Cite_book.
- Alternating_knot wikiPageUsesTemplate Template:Knot_theory.
- Alternating_knot wikiPageUsesTemplate Template:Main.
- Alternating_knot wikiPageUsesTemplate Template:MathWorld.
- Alternating_knot wikiPageUsesTemplate Template:Reflist.
- Alternating_knot subject Category:Alternating_knots_and_links.
- Alternating_knot subject Category:Knot_invariants.
- Alternating_knot type Invariant.
- Alternating_knot type Concept.
- Alternating_knot comment "In knot theory, a knot or link diagram is alternating if the crossings alternate under, over, under, over, as one travels along each component of the link. A link is alternating if it has an alternating diagram.Many of the knots with crossing number less than 10 are alternating. This fact and useful properties of alternating knots, such as the Tait conjectures, was what enabled early knot tabulators, such as Tait, to construct tables with relatively few mistakes or omissions.".
- Alternating_knot label "Alternating knot".
- Alternating_knot sameAs Alternierendes_Knotendiagramm.
- Alternating_knot sameAs 交代結び目.
- Alternating_knot sameAs m.04kzvb.
- Alternating_knot sameAs Q4736417.
- Alternating_knot sameAs Q4736417.
- Alternating_knot wasDerivedFrom Alternating_knot?oldid=681268186.
- Alternating_knot depiction Knot_8sb19.svg.
- Alternating_knot isPrimaryTopicOf Alternating_knot.