Hyperbolic structures on link complements, octahedral decompositions, and quantum $\mathfrak{sl}_2$

Abstract

Blanchet et al., "Holonomy braidings, biquandles and quantum invariants of links with SL 2 (C) at connections". arXiv [KKY18] H. Kim, S. Kim, and S. Yoon, "Octahedral developing of knot complement. I: Pseudo-hyperbolic structure". arXiv C Previously Blanchet et al. [Bla+20] have shown versions of 1 and 3, and Kim, Kim, and Yoon [KKY18] have shown versions of and 1 and 2. Closely related to their work is a description of boundary-parabolic SL 2 (C)-structures for braid closures in terms of cluster variables due to Hikami and Inoue [HI15].is paper improves these results and places them in a uni ed context. Our key idea is to consider a presentation of U g (sl 2 ) in terms of a quantum cluster algebra. is presentation leads to cluster-type coordinates on the SL 2 (C)-representation variety of a tangle complement, which are in turn naturally related to the octahedral decomposition. Plan of the paper• In the remainder of the introduction we give more background on the hyperbolic geometry of 3-manifolds and connections to quantum topology.• In Section 2 we de ne shaped tangle diagrams and explain how they relate to SL 2 (C)structures. We give an algebraic proof that, for any diagram D of a link L, up to conjugacy every SL 2 (C)-structure on L is detected by a shaping of D. We also give some examples of shaped link diagrams.• In Section 3 we relate shaped tangle diagrams to geometry using the octahedral decomposition. We use this perspective to strengthen our existence result: we can always nd geometrically nondegenerate shapings for any nontrivial SL 2 (C)-structure.• In Section 4 we explain the connection to quantum groups and cluster algebras. is perspective allows us to interpret our results as a stronger version of the Hikami-Inoue conjecture [HI15] and its solution by Cho, Yoon, and Zickert [CYZ20].