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Compute oriented linking number from a diagram
Definition
Assume .
The input is a finite regular oriented link diagram together with its declared
components. If it does not have exactly two components, return
invalid-input. For a two-component diagram , initialize and make
one pass over its finite crossing list. At each crossing whose branches lie on
different components, add
to , where the determinant is taken in the oriented projection plane; ignore
self-crossings. Return . The even-sum lemma establishes that this output is
an integer and is the diagram's linking number.
Depends on
Used by
- Pairwise linking numbers do not classify labeled links Counterexample
- Framed knots, self-linking, and the Seifert framing Definition
- Hopf and Whitehead link linking-number computations Example
- The Borromean string link has nonzero triple Milnor invariant Example
- The degree-one derivative of linking number Example
- Small framed push-offs define a well-defined self-linking number Lemma
- The one-term relation and framing Lemma
- Blackboard self-linking equals diagram self-writhe Theorem
- Linking number and Fox coloring finite complexity Theorem
- Linking number equals signed intersection with a spanning surface Theorem
- Linking number is invariant and writhe changes under a kink Theorem
- The diagram linking-number scan terminates and returns the stated integer Theorem
Dependency tree · two levels
24 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, University of Chicago course notes (2025), §3.3, pp. 26–29 (standard reference, not scraped)
- Mohamed Elhamdadi, Mustafa Hajij, and Kyle Istvan, Framed Knots, arXiv:1910.10257v6 (2020), §3, pp. 3–9 (standard reference, not scraped)