DefinitionDefinition: Literature-sourcedProof: AI-adaptedPipeline-generatedjudge pass (gpt-6-luna)verified 2026-10-22 (gpt-6.1-sol)

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Crossing number of a link

Definition

Assume ${A C}_{ω}$ . Let $K$ be an unoriented link type in $S^{3}$ . Its crossing number $c (K)$ is the least element of ${n \in N : some regular diagram of K has exactly n crossings} .$

Definition references

Calegari, Introduction to Knot Theory, §1.9

Menasco and Thistlethwaite, The classification of alternating links (1993)

Facts & Assumptions

Given: ${A C}_{ω}$ and an unoriented link type $K$ in $S^{3}$ .

[F1]

A link is a finite disjoint union of smoothly embedded circles in $S^{3}$ , and it may be moved off $\infty$ into $R^{3}$ before projection (Oriented links in the three-sphere and ambient isotopy).

[F2]

A regular diagram is a finite planar projection with transverse double points, no triple points, and over-under data (Regular oriented link diagrams).

[F3]

Under countable choice, every oriented link admits a regular projection (Existence of regular projections).

[F4]

Every nonempty subset of $N$ has a least element (The well-ordering principle).

[A1]

Countable choice, abbreviated ${A C}_{ω}$ , selects one element from each member of a countable family of nonempty sets (The Axiom of Countable Choice ( ${A C}_{ω}$ )).

Proof

1.1givenF1F2F3A1construct

Let $N (K)$ be the displayed set of crossing counts. By [F1], move an oriented representative off $\infty$ into $R^{3}$ ; [F3] and [A1] give it a regular diagram. Forgetting the diagram's strand arrows gives a regular diagram of the same unoriented type, so $N (K) \neq \emptyset$ .

2.1givenF1F4step 1.1choose∎

The set $N (K)$ is a nonempty subset of $N$ , so [F4] supplies its unique least element. Ambient-isotopic representatives have the same crossing-count set by definition of the underlying unoriented link type. This makes $c (K)$ well-defined.

Depends on

Used by

Jones span detects crossing number for reduced alternating knots Corollary
A nugatory kink defeats crossing minimality Counterexample
The unknot defeats a universal coefficient-bound extension Counterexample
A trefoil Jones–Vassiliev coefficient calculation Example
Trefoil and figure-eight Jones spans Example
Jones polynomial unknot-detection status Remark
Reduced alternating diagrams minimize crossing number Theorem

Dependency tree · two levels

27 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, §1.9, printed pp. 7–8 (standard reference, not scraped)
William W. Menasco and Morwen B. Thistlethwaite, The classification of alternating links, Annals of Mathematics 138 (1993), pp. 113–171 (standard reference, not scraped)