The ribbon cocycle invariant is defined by means of a partition function using ternary cohomology of self-distributive structures (TSD) and colorings of ribbon diagrams of a framed link, following the same paradigm introduced by Carter, Jelsovsky, Kamada, Langford and Saito in Transactions of the American Mathematical Society 2003;355(10):3947-89, for the quandle cocycle invariant. In this article we show that the ribbon cocycle invariant is a quantum invariant. We do so by constructing a ribbon category from a TSD set whose twisting and braiding morphisms entail a given TSD $2$-cocycle. Then we show that the quantum invariant naturally associated to this braided category coincides with the cocycle invariant. We generalize this construction to symmetric monoidal categories and provide classes of examples obtained from Hopf monoids and Lie algebras. We further introduce examples from Hopf-Frobenius algebras, objects studied in quantum computing.
This research has been funded by the Estonian Research Council under the grant: MOBJD679. The author is grateful to M. Elhamdadi and M. Saito for useful conversations, as well as to the referee for useful suggestions that have improved the quality of the article.
"Quantum invariants of framed links from ternary self-distributive cohomology." Osaka J. Math. 59 (4) 777 - 820, October 2022.