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Reduced alternating link diagram
Definition
A regular link diagram is connected when its underlying projection graph is connected. It is alternating when successive crossing visits along each component switch between overpasses and underpasses. It is reduced when no complementary region of the projection sphere has a crossing more than once on its boundary. A connected alternating diagram that is reduced is a reduced alternating link diagram.
Definition references
Facts & Assumptions
Given: A regular link diagram with its finite projection graph and over-under data.
A regular oriented link diagram has finitely many transverse double points, no triple points, and an over-under datum at each crossing (Regular oriented link diagrams).
Proof
The projection graph, its complementary regions, and the cyclic order of crossing visits along each component are finite data by [F1]. Thus connectedness, alternation, and the stated repeated-region condition are properties determined by the given diagram.
Forgetting all strand orientations preserves the projection graph and the over-under data, so each condition is unchanged by reversing component orientations. The definition therefore applies consistently to the underlying unoriented diagram.
Depends on
Used by
- Jones span detects crossing number for reduced alternating knots Corollary
- A nugatory kink defeats crossing minimality Counterexample
- Trefoil and figure-eight Jones spans Example
- Adequacy and Jones span of reduced alternating diagrams Lemma
- Connected alternating diagrams represent nonsplit links Lemma
- Connected-diagram bracket span bound and alternating equality Lemma
- The checkerboard normal pairing of a connected alternating diagram is definite Lemma
- Reduced alternating diagrams minimize crossing number Theorem
- Tait flyping theorem Theorem
Dependency tree · two levels
4 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.
Sources
- Danny Calegari, Introduction to Knot Theory, Definition 5.17, printed pp. 43–44 (standard reference, not scraped)
- Thomas Kindred, A geometric proof of the flyping theorem, §2.1, arXiv:2008.06490v3 (standard reference, not scraped)