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Adequacy and Jones span of reduced alternating diagrams

Statement

Assume full AC and let $D$ be a nonempty connected reduced alternating diagram with $c$ crossings. At every crossing, changing one smoothing from the all- $A$ state or from the all- $B$ state merges two distinct state circles; these are A- and B-adequacy. The Kauffman bracket has $A$ -span $4 c$ . If $D$ represents a knot $K$ , then the library Jones polynomial has $t$ -span $c$ , and every $c^{'}$ -crossing diagram of $K$ has $t$ -span at most $c^{'}$ under the convention $t = A^{- 4}$ . The span of a nonzero Laurent polynomial is its largest exponent minus its smallest exponent.

Statement references

Calegari, Introduction to Knot Theory, §5.2

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

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

Facts & Assumptions

Given: Full AC and a nonempty connected reduced alternating diagram $D$ with $c$ crossings.

[F1]

In a reduced alternating diagram, successive crossing visits along each component switch over-under, and no complementary region has a crossing more than once on its boundary (Reduced alternating link diagram).

[F2]

For a nonempty connected reduced alternating diagram, each one-crossing change from either extreme state merges distinct circles, both extreme coefficients are nonzero, and the bracket span is $4 c$ ; for any nonempty connected diagram its bracket span is at most $4 c$ (Connected-diagram bracket span bound and alternating equality).

[F3]

Under full AC, the library Jones polynomial satisfies $V_{K} (t) = f_{K} (t^{- 1 / 4})$ , where $f_{D} (A) = (- A^{3})^{- wr (D)} ⟨ D ⟩$ and $t = A^{- 4}$ (The bracket state sum matches the library Jones specialization).

[A1]

The Axiom of Choice supplies a choice function for every family of nonempty sets (The Axiom of Choice).

Proof

1.1givenF1F2

By [F1], the input is a connected reduced alternating diagram, so the finite state-merging conclusion in [F2] applies. It establishes both A-adequacy and B-adequacy without invoking the separate Jones axiom branch, and gives ${span}_{A} ⟨ D ⟩ = 4 c$ .

2.1A1F2F3step 1.1givenalgebra∎

Suppose $D$ represents a knot $K$ . Let $E$ be any $c^{'}$ -crossing diagram of the same knot, with its component oriented to represent the chosen orientation of $K$ . The projection of a one-component diagram is connected because it is the continuous image of a circle; thus [F2] gives ${span}_{A} ⟨ E ⟩ \leq 4 c^{'}$ . Multiplication by the writhe-normalizing monomial $(- A^{3})^{- wr (E)}$ shifts all exponents equally and does not change span. By [A1] and [F3], substituting $t = A^{- 4}$ divides the exponent span by $4$ , so ${span}_{t} V_{K} (t) \leq c^{'}$ . Applying this to $D$ and using step 1.1 gives ${span}_{t} V_{K} (t) = c$ .

Depends on

Used by

Jones span detects crossing number for reduced alternating knots Corollary

Dependency tree · two levels

23 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, §5.2, printed pp. 42–45 (standard reference, not scraped)
Louis H. Kauffman, New Invariants in the Theory of Knots, Astérisque 163–164 (1988), §III, printed pp. 160–164 (standard reference, not scraped)
Louis H. Kauffman, Combinatorial Knot Theory and the Jones Polynomial, arXiv:2204.12104v2, §2 (standard reference, not scraped)