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A nugatory kink defeats crossing minimality

Statement refuted

Assume full AC. The positive one-crossing type-I kink of the unknot is connected and alternating but not reduced; its diagram has one crossing while the represented unknot has crossing number zero, and its normalized library Jones polynomial has $t$ -span zero.

The arrows orient the single component; at the central crossing, the diagonal from northwest to southeast is over and the southwest-to-northeast diagonal has a gap.

Statement references

Calegari, Introduction to Knot Theory, §5.2

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

Facts & Assumptions

Given: Full AC, the positive type-I kink diagram $D_{+}$ of the unknot, and the bracket and Jones conventions in the statement.

[F1]

A crossing contributes $A$ for its $A$ -smoothing and $A^{- 1}$ for its $B$ -smoothing, a state with $k$ circles contributes $δ^{k - 1}$ , and $δ = - A^{2} - A^{- 2}$ (Kauffman bracket smoothing and loop value).

[F2]

The positive geometric crossing has sign $+ 1$ , and writhe is the sum of crossing signs (Oriented crossing signs, diagram writhe, and blackboard framing).

[F3]

A connected diagram has connected projection graph, alternation switches over-under at successive visits, and reducedness excludes a crossing repeated on a region boundary (Reduced alternating link diagram).

[F4]

A type-I Reidemeister move is realized by an ambient isotopy of the represented link (Each oriented Reidemeister move is realized by an ambient isotopy).

[F5]

Under full AC, the normalized bracket state sum specializes to the library Jones polynomial by $V_{K} (t) = f_{K} (t^{- 1 / 4})$ with $t = A^{- 4}$ (The bracket state sum matches the library Jones specialization).

[F6]

The crossing number is the least crossing count among regular diagrams of the same underlying unoriented link type (Crossing number of a link).

[A1]

Full AC is the choice-function principle for arbitrary families of nonempty sets (The Axiom of Choice).

Counterexample

technique · direct

1.1givenF1constructalgebra

The two states of a positive kink consist of an $A$ -smoothing that leaves the original arc and creates one trivial circle, and a $B$ -smoothing that leaves only the original arc. By [F1] their total bracket factor is $A δ + A^{- 1} = A (- A^{2} - A^{- 2}) + A^{- 1} = - A^{3}$ , so $⟨ D_{+} ⟩ = - A^{3}$ .

2.1A1F2F3F5step 1.1algebra

The kink has one positive crossing, so [F2] gives $wr (D_{+}) = 1$ . Its normalized bracket is $f_{D_{+}} (A) = (- A^{3})^{- 1} (- A^{3}) = 1$ . By [A1] and [F5], the library Jones polynomial is therefore $1$ , whose $t$ -span is $0$ . The one-crossing projection is connected; its unique component visits the crossing once over and once under, so it is alternating; the exterior region has the crossing twice on its boundary, so it is not reduced by [F3].

3.1F4F6givenstep 2.1algebra∎

By [F4], the type-I move carries $D_{+}$ to the crossing-free unknot diagram. Thus the represented link has a diagram with zero crossings. Since crossing counts are nonnegative by [F6], its crossing number is zero, strictly less than the one crossing in $D_{+}$ . This verifies the claimed failure of crossing minimality at a non-reduced alternating diagram.

Depends on

Used by

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Sources

Danny Calegari, Introduction to Knot Theory, §5.2, printed pp. 43–45 (standard reference, not scraped)
Louis H. Kauffman, New Invariants in the Theory of Knots, Astérisque 163–164 (1988), §III, printed pp. 164–168 (standard reference, not scraped)