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

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Bracket smoothings of positive and negative braid crossings

Statement

Let $σ_{i}$ be the positive geometric Artin braid crossing in a braid rectangle whose parameter increases from bottom to top, with strand $i$ passing below and strand $i + 1$ passing above. Use the bracket chart in which the over-strand runs southwest to northeast, the $A$ -smoothing joins left to left and right to right with weight $A$ , and the $B$ -smoothing joins top to top and bottom to bottom with weight $A^{- 1}$ . Let $I$ be the through pairing and let $e_{i}$ be the pairing with a top cap and a bottom cup at positions $i, i + 1$ . Then the positive crossing has bracket operator $A I + A^{- 1} e_{i}$ , and the negative crossing has bracket operator $A^{- 1} I + A e_{i}$ .

Statement references

Kauffman, New Invariants in the Theory of Knots, §II

Kauffman, Combinatorial Knot Theory and the Jones Polynomial, §2

Facts & Assumptions

Given: The positive geometric half twist, its inverse, the determinant crossing-sign convention, and the Kauffman bracket chart specified in the statement.

[F1]

In the positive geometric generator, strand $i$ moves from $x = - h$ to $x = h$ below the midpoint and strand $i + 1$ moves from $x = h$ to $x = - h$ above it; the inverse reverses the crossing (The elementary geometric half twist, its support disc, and its opposite).

[F2]

The crossing sign is $sgn \det (v_{o v e r}, v_{u n d e r})$ , and this convention assigns sign $+ 1$ to the positive geometric Artin half twist (Oriented crossing signs, diagram writhe, and blackboard framing).

[F3]

In the stated chart a crossing contributes $A$ times its $A$ -smoothing plus $A^{- 1}$ times its $B$ -smoothing; the loop value is $δ = - A^{2} - A^{- 2}$ (Kauffman bracket smoothing and loop value).

Proof

1.1F1F2F3construct

Use source coordinates $(u, v)$ with their standard positive orientation and positive projection coordinates $(t, x)$ on the braid box. For $h > 0$ , the map $(t, x) = (v, - 2 h u)$ has determinant $2 h > 0$ . It sends the source over direction $(1, 1)$ to $(1, - 2 h)$ and the source under direction $(- 1, 1)$ to $(1, 2 h)$ , whose determinant is $4 h > 0$ . The source left/right pairs have constant $u$ , hence constant braid position $x$ and give the through pairing $I$ ; the source top/bottom pairs have constant $v$ , hence constant braid time $t$ and give the cup-cap $e_{i}$ . The displayed page coordinates $(x, t)$ reverse the positive projection orientation; their visual quarter-turn is not the orientation comparison. The two terms in [F3] are therefore $A I$ and $A^{- 1} e_{i}$ .

2.1F1F3step 1.1construct∎

The inverse half twist reverses which diagonal is over. Reflection of the local crossing interchanges the two pairings while retaining the bracket rule that assigns coefficient $A$ to the chart's $A$ -smoothing and $A^{- 1}$ to its $B$ -smoothing. Consequently the through pairing now has coefficient $A^{- 1}$ and the cap-cup pairing has coefficient $A$ .

Depends on

Used by

A trefoil Jones–Vassiliev coefficient calculation Example
Trefoil and figure-eight Jones spans Example

Dependency tree · two levels

14 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

Louis H. Kauffman, New Invariants in the Theory of Knots, Astérisque 163–164 (1988), §II, pp. 155–160 (standard reference, not scraped)
Louis H. Kauffman, Combinatorial Knot Theory and the Jones Polynomial, arXiv:2204.12104v2, §2 (standard reference, not scraped)