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The ribbon evaluation is an invariant of framed colored links

Statement

Assume ${A C}_{ω}$ (The Axiom of Countable Choice ( ${A C}_{ω}$ )). Let $C$ be a ribbon category and $X \in C$ . If $β \in B_{n}$ and $β^{'} \in B_{n^{'}}$ , with $n, n^{'} \geq 1$ , are braids whose blackboard-framed $X$ -colored closures are isotopic as framed oriented tangles, then

$t_{n} (β) = t_{n^{'}} (β^{'})$

in ${End}_{C} (1)$ . More generally, $t$ is an invariant of framed $X$ -colored links: the value $t_{n} (β)$ depends only on the framed isotopy class of the closure of $β$ . Ordinary Markov stabilization is not a framed isotopy: it inserts a curl and is handled only after writhe normalization.

Facts & Assumptions

Given: ${A C}_{ω}$ ; a ribbon category $C$ , an object $X$ , braids $β \in B_{n}$ and $β^{'} \in B_{n^{'}}$ whose blackboard-framed $X$ -colored closures are isotopic as framed oriented tangles.

[L1]

The ribbon evaluation equals the evaluation of the blackboard-framed closure: $t_{n} (β) = F_{X} ({\hat{β}}^{f r})$ , and the same for $β^{'}$ ; moreover the closure of the stabilized braid $ι_{n} (β) σ_{n}^{\pm 1}$ is the closure of $β$ with one full twist added on a band (The ribbon trace equals the framed-closure evaluation).

[L2]

The tangle evaluation functor assigns equal values to isotopic framed tangles: isotopic framed tangles are equal morphisms of the framed oriented tangle category, and a functor preserves equalities (A ribbon object defines a unique framed-tangle evaluation functor).

Proof

technique · direct

1.1L1L2given

The two evaluations agree. By [L1] $t_{n} (β) = F_{X} ({\hat{β}}^{f r})$ and $t_{n^{'}} (β^{'}) = F_{X} ({\hat{β^{'}}}^{f r})$ . The closures are isotopic framed oriented tangles by hypothesis, so by [L2] their values under $F_{X}$ are equal. Hence $t_{n} (β) = t_{n^{'}} (β^{'})$ .

1.2L1given

The framing changes under stabilization. By [L1], stabilization inserts one signed full twist. A framing number is the linking number of a component with its normal push-off; a full twist changes that number by $\pm 1$ (Turaev, Chapter I §2.1, printed pp. 34--35). The sum of component framing numbers is preserved by framed isotopy, including permutation of components, and changes by $\pm 1$ here. Thus stabilization is not a framed isotopy. Its evaluations can nevertheless coincide in a particular category; the functorial invariance alone gives no stabilization identity.

2.1L1L2step 1.1construct

Framed-link invariance. On any closed framed $X$ -colored tangle $L$ , define its evaluation to be $F_{X} (L)$ . By [L2] this is a framed-isotopy invariant, and [L1] identifies it with $t_{n} (β)$ whenever $L$ is the blackboard-framed closure of $β$ . On the empty tangle the strict-model evaluator is the identity of the unit, agreeing with $t_{0} (e)$ . This constructs the general evaluation without assuming that every framing has a blackboard-braid representative.

3.1step 1.1step 2.1step 1.2∎

Conclusion. Steps 1.1--1.2 give the framed-isotopy invariance of the ribbon evaluation, and step 1.2 records that ordinary Markov stabilization changes the framing and lies outside this statement. The only choice principle used is ${A C}_{ω}$ , consumed through the closure comparison of [L1] and the functor $F_{X}$ of [L2]; the final comparison of the two values is then functoriality of $F_{X}$ applied to equal morphisms.

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Sources

V. G. Turaev, Quantum Invariants of Knots and 3-Manifolds (de Gruyter Studies in Mathematics 18, 1994) (standard reference, not scraped)
P. Etingof, S. Gelaki, D. Nikshych, and V. Ostrik, Tensor Categories (AMS Mathematical Surveys and Monographs 205), author's final version (standard reference, not scraped)