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Colored framed-link evaluation from a ribbon category
Definition
Assume the Axiom of Countable Choice. Let be a ribbon category with chosen left duals and tensor unit . A finite oriented framed link colored in is a finite disjoint union of oriented smoothly embedded circles in , a homotopy class of nowhere-zero normal vector field on each component, and an assignment of an object of to each component. A framed colored isotopy carries orientations, framings, and component colors. If is the ribbon-tangle evaluation functor, define This value depends only on the framed colored isotopy class. It is not automatically a field scalar; if is -linear and , it is a scalar in . The empty link is assigned .
Definition references
Facts & Assumptions
Given: The ribbon category, its chosen left duals, a finite oriented framed link, and an object color on every component.
For arbitrary component colors, the ribbon evaluation assigns a morphism to each colored framed tangle, is invariant under framed graph isotopy, and preserves the empty graph (Multicolored ribbon graphs with arbitrary coupons have a unique evaluation).
Countable choice means every sequence of nonempty sets has a choice function (The Axiom of Countable Choice ()).
Proof
Given: The data in the Definition and [A1].
Under [A1], the evaluator in [F1] exists. The colored framed link is a ribbon graph with no boundary endpoints, so it assigns a morphism from the tensor unit to itself. This is an element of , with no field-valued conclusion unless that endomorphism ring has been identified with a field.
If two representatives are framed colored isotopic, [F1] gives equal evaluator morphisms; the empty graph has value by the same fact. Therefore the displayed assignment is well-defined on framed colored isotopy classes and has the stated empty-link value.
Depends on
Used by
- A decomposable color with nonscalar twist has no single writhe correction Counterexample
- Normalized colored Jones sequence Definition
- The normalized generic colored Jones quotient is an integral Laurent polynomial Lemma
- Finite colored Jones evaluation is exact and representative-independent Theorem
- Framed colored-braid representative independence Theorem
- Scalar twist correction removes the framing Theorem
Dependency tree · two levels
18 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
- 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, §§8.9–8.10 (standard reference, not scraped)