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Framed colored-braid representative independence
Statement
Assume countable choice. Let be a ribbon category with chosen left duals. If two finite colored braids have color-preserving endpoint permutations and their blackboard-framed closures are isotopic as oriented framed links by an isotopy preserving each component color, then their evaluations agree in . Thus the framed colored evaluation of a closed braid is independent of its colored-braid representative whenever the represented framed colored links are the same.
Statement references
Facts & Assumptions
Given: Countable choice, a ribbon category, two finite colored braids, and a framed colored isotopy between their closures.
The colored framed-link evaluation is the framed ribbon-tangle functor applied to the closed link and takes values in (Colored framed-link evaluation from a ribbon category).
The multicolored ribbon-tangle evaluator is invariant under every boundary-relative framed graph isotopy and handles arbitrary object colors with the chosen duality maps (Multicolored ribbon graphs with arbitrary coupons have a unique evaluation).
Countable choice is assumed for the cited construction of the multicolored tangle evaluator (The Axiom of Countable Choice ()).
Proof
Given: The data in the Statement and [A1].
Under [A1], each braid closure is a framed colored link with empty boundary. By [F1], its value is the endomorphism of the tensor unit assigned by the multicolored framed-tangle evaluator, and its component labels are exactly the colors carried around the permutation cycles.
The isotopy hypothesis is a framed graph isotopy that preserves the labels. By [F2], the two closed tangles have equal evaluation morphisms. Applying [F1] identifies those morphisms with the two braid-closure values, proving equality in . This statement concerns framed isotopy; an ordinary Markov stabilization that changes framing is not an additional move here.
Depends on
Used by
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Dependency tree · two levels
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Sources
- V. G. Turaev, Quantum Invariants of Knots and 3-Manifolds, Chapter I, Theorem 2.5 and §§3–4 (standard reference, not scraped)
- P. Etingof, S. Gelaki, D. Nikshych, and V. Ostrik, Tensor Categories, §8.10 (standard reference, not scraped)