TheoremStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedverified 2026-10-22 (gpt-6.1-sol)

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Blackboard self-linking equals diagram self-writhe

Statement

Assume the Axiom of Countable Choice. Let $D$ be a regular oriented diagram of a knot $K$ with blackboard framing $b_{D}$ . The self-linking number of the framed knot $(K, b_{D})$ equals the diagram writhe: $sl (K, b_{D}) = wr (D) .$ More generally, let $D$ be a regular oriented diagram of an oriented link $L = K_{1} \cup \dots \cup K_{r}$ , and let $b_{D, i}$ be the blackboard framing of $K_{i}$ induced by its component subdiagram. Then $sl (K_{i}, b_{D, i}) = {wr}_{i} (D) (1 \leq i \leq r) .$ Total writhe of a link still includes the signs of mixed crossings, while the sum of component self-linking numbers uses only self-crossings.

Statement references

Regular oriented diagrams

Crossing signs, writhe, and blackboard framing

Crossing count for a blackboard push-off

Self-linking of a framed knot

Countable-choice hypothesis

Facts & Assumptions

Given: The Axiom of Countable Choice and a regular oriented diagram $D$ of a knot $K$ , or, for the component claim, of a link $L = K_{1} \cup \dots \cup K_{r}$ .

[A1]

The Axiom of Countable Choice, written ${A C}_{ω}$ (The Axiom of Countable Choice ( ${A C}_{ω}$ )).

[F1]

The self-linking number of a framed knot is the linking number with a sufficiently small embedded push-off along its framing section (Small framed push-offs define a well-defined self-linking number, Framed knots, self-linking, and the Seifert framing).

[F2]

For a two-component diagram the algorithmic linking number is one half of the signed sum of its mixed crossing signs (Compute oriented linking number from a diagram, The diagram linking-number scan terminates and returns the stated integer).

[F3]

The blackboard push-off has exactly two mixed crossings of the original sign for each original self-crossing and no other mixed crossings (A blackboard push-off doubles each crossing into mixed crossings).

Proof

1.1A1F3givenalgebra

By [F3], the mixed crossing list for $K \cup K_{b_{D}}$ has two crossings of sign $sgn (c)$ for each crossing $c$ of $D$ and no others. Therefore its signed mixed sum is $2 \sum_{c} sgn (c) = 2 wr (D)$ .

2.1A1F1F2step 1.1construct∎

By [F2], the linking number of $K$ with its blackboard push-off is half this mixed sum; by [F1] it is the self-linking number. Thus $sl (K, b_{D}) = wr (D)$ . For a link diagram $D$ and each component $K_{i}$ , apply the same calculation to its regular component subdiagram with induced blackboard framing $b_{D, i}$ ; crossings with other components are not mixed crossings of that component and its push-off, so precisely its self-crossings contribute.

Depends on

Used by

Writhe is not an invariant of an unframed knot Counterexample
Group presentation for positive trefoil +1 surgery Lemma
The one-term relation and framing Lemma
Linking number is invariant and writhe changes under a kink Theorem
Rack colorings are framed Reidemeister invariants Theorem

Dependency tree · two levels

38 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

Mohamed Elhamdadi, Mustafa Hajij, and Kyle Istvan, Framed Knots, arXiv:1910.10257v6 (2020), §3, pp. 3–9 (standard reference, not scraped)
Danny Calegari, Introduction to Knot Theory, University of Chicago course notes (2025), §3.3, pp. 26–29 (standard reference, not scraped)