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

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Kauffman bracket smoothing and loop value

Definition

Let $A$ be an indeterminate and let $Z [A^{\pm 1}]$ be the Laurent polynomial ring in $A$ . A nonempty unoriented regular link diagram is the finite planar projection and over/under data of a regular oriented link diagram after its strand arrows are forgotten. Its finite crossing set is denoted $C (D)$ , and at least one closed component is required. The empty link diagram is outside this definition.

At a crossing use the following local chart: the over-strand runs from the southwest endpoint to the northeast endpoint and is drawn continuously over the under-strand, which runs from northwest to southeast with a gap at the intersection. Horizontal-right followed by vertical-up is the positive orientation of this displayed chart. In this chart the A-smoothing joins the two left endpoints and joins the two right endpoints; the B-smoothing joins the two top endpoints and joins the two bottom endpoints. These labels are attached to this displayed chart and are transported by orientation-preserving changes of local coordinates. Thus turning right from an entering underpassing half-strand gives the A-smoothing.

For a state $S : C (D) \to {A, B}$ , let $a (S)$ and $b (S)$ be the numbers of A- and B-smoothings, and let $∣ S ∣$ be the number of closed circles in the crossingless diagram obtained by performing all smoothings. Set $δ : = - A^{2} - A^{- 2}$ and define the Kauffman bracket of $D$ by $⟨ D ⟩ : = \sum_{S : C (D) \to {A, B}} A^{a (S) - b (S)} δ^{∣ S ∣ - 1} \in Z [A^{\pm 1}] .$ Because $D$ is nonempty, every complete smoothing has $∣ S ∣ \geq 1$ , so the loop exponent is nonnegative. A crossingless $k$ -component unlink with $k \geq 1$ has bracket $δ^{k - 1}$ .

Definition references

Regular oriented link diagrams

Planar isotopy of link diagrams

Kauffman, An Invariant of Regular Isotopy (1990), §III, printed pp. 430–432

Kauffman, Combinatorial Knot Theory and the Jones Polynomial (2022), §2

Roberts, Knots Knotes (1999), §§4.1 and 4.3

Funar, A Crash Course in Quantum Invariants of Knots and 3-Manifolds, Chapter 3, §1.1

Facts & Assumptions

Given: A nonempty unoriented regular link diagram $D$ with finite crossing set $C (D)$ .

[F1]

A regular oriented link diagram has finitely many transverse double points, no triple points, and an over/under datum at every crossing; forgetting the strand arrows leaves the input used here (Regular oriented link diagrams).

[F2]

A planar isotopy is an orientation-preserving smooth family of planar diffeomorphisms that transports the diagram and its over/under data and creates or deletes no crossings (Planar isotopy of link diagrams).

Proof

1.1givenF1construct

Since $C (D)$ is finite by [F1], the set of all functions $S : C (D) \to {A, B}$ is finite with exactly $2^{∣ C (D) ∣}$ elements. Each state replaces every crossing in a finite collection of disjoint crossing disks by one of the two specified pairings. The resulting finite 1-manifold has at least one circle because the input diagram has at least one component. Hence each term $A^{a (S) - b (S)} δ^{∣ S ∣ - 1}$ is a defined Laurent polynomial.

2.1givenF2step 1.1algebra∎

The displayed sum is indexed by the set of states itself, so reordering a list of crossings merely gives a bijective reindexing of the same finite sum. By [F2], a planar isotopy carries crossings, local charts, states, and resulting circles bijectively to those of the isotoped diagram, preserving $a (S)$ , $b (S)$ , and $∣ S ∣$ . Therefore the formula is well-defined on planar-isotopy classes and is choice-free.

Depends on

Used by

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Sources

Louis H. Kauffman, An Invariant of Regular Isotopy, Transactions of the AMS 318(2) (1990), §III, printed pp. 430–432 (standard reference, not scraped)
Louis H. Kauffman, Combinatorial Knot Theory and the Jones Polynomial, arXiv:2204.12104v2, §2 (standard reference, not scraped)
Justin Roberts, Knots Knotes, §§4.1 and 4.3, printed pp. 21 and 24–25 (standard reference, not scraped)
François Funar, A Crash Course in Quantum Invariants of Knots and 3-Manifolds, Chapter 3, §1.1 (standard reference, not scraped)