Ribbon evaluation from a colored braid word

An effective field is a field whose elements have finite encodings and whose exact addition, subtraction, multiplication, equality test, and inversion of nonzero elements are terminating procedures. Input a finite-dimensional strict matrix realization over an effective field $k$ of a $k$-linear ribbon category with $\mathrm{End}(\mathbf{1})=k$, finite-dimensional color objects with fixed bases, exact matrices for every braiding and inverse braiding used, the evaluation and coevaluation maps for the closure, the twist matrices, and the matrix $J_W$ of the ribbon-induced pivotal comparison $j_W=u_W\circ {\theta }_W:W\to W^{\vee \vee }$ for $W=V_{a_1}\otimes \cdots \otimes V_{a_n}$, where $u_W$ is the Drinfeld morphism. Also input a finite braid word $\beta$ on $n\ge 1$ strands and the top color tuple $(a_1,\dots ,a_n)$. The matrices are assumed to satisfy the stated ribbon-category axioms; an implementation does not infer those axioms from arbitrary matrices.

Use the strand-position permutation homomorphism $\pi :B_n\to \mathrm{Sym}(\{1,\dots ,n\})$ given by $\pi ({\sigma }_i)=(ii+1)$. Read the braid letters in composition order, updating the current color tuple at each crossing by swapping its adjacent colors. For a positive letter ${\sigma }_i$, use the matrix of $1\otimes \cdots \otimes c_{V_{a_i},V_{a_{i+1}}}\otimes \cdots \otimes 1,$ and for a negative letter ${\sigma }_i^{-1}$ use $1\otimes \cdots \otimes c_{V_{a_{i+1}},V_{a_i}}^{-1}\otimes \cdots \otimes 1$, whose source adjacent factors are $V_{a_i}\otimes V_{a_{i+1}}$. Here the subscripts always refer to the current tuple before the swap. Let the blackboard-framed closure join the bottom and top endpoint at each position by the identity pairing. Reject the input if the final color tuple differs from the initial tuple: the tuple records the colors at the paired endpoints, so equality is exactly the condition that the closure carries a consistent color on every component. Starting with the identity matrix, multiply the typed crossing matrices in word order to obtain $M_{\beta }:W\to W$, then return the finite matrix contraction ${\mathrm{ev}}_{W^{\vee }}((J_W{M}_{\beta })\otimes 1_{W^{\vee }}){\mathrm{coev}}_W\in k\mathrm{.}$ In ordinary dual-basis coordinates this contraction is the matrix trace $\mathrm{Tr}(J_W{M}_{\beta })$. No algorithm is asserted for arbitrary abstract ribbon categories without effective finite matrix data.

Correctness and termination of this procedure ↗

Colored framed-link value

Mixed-color ribbon-tangle evaluation

Categorical trace contraction

Drinfeld morphism and its tensor relation

Balancing and ribbon dual compatibility

Duality maps and snake identities

Artin braid words

Permutations of the strand labels

Turaev, Quantum Invariants of Knots and 3-Manifolds, Chapter I, Theorem 2.5 and §§3–4

Etingof, Gelaki, Nikshych, and Ostrik, Tensor Categories, §§8.9–8.10