In order to apply quantum topology methods to nonplanar graphs, we define a planar diagram category that describes the local topology of embeddings of graphs into surfaces. These virtual graphs are a categorical interpretation of ribbon graphs. We describe an extension of the flow polynomial to virtual graphs, the S-polynomial, and formulate the sl(N ) Penrose polynomial for non-cubic graphs, giving contraction-deletion relations. The S-polynomial is used to define an extension of the Yamada polynomial to virtual spatial graphs, and with it we obtain a sufficient condition for non-classicality of virtual spatial graphs. We conjecture the existence of local relations for the S-polynomial at squares of integers.

We show that the reduced SL 2 .C/-twisted Burau representation can be obtained from the quantum group U q .sl 2 / for q D i a fourth root of unity and that representations of U q .sl 2 / satisfy a type of Schur-Weyl duality with the Burau representation. As a consequence, the SL 2 .C/-twisted Reidemeister torsion of links can be obtained as a quantum invariant. Our construction is closely related to the quantum holonomy invariant of Blanchet-Geer-Patureau-Mirand-Reshetikhin, and we interpret their invariant as a twisted Conway potential.

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].

Kashaev and Reshetikhin previously described a way to define holonomy invariants of knots using quantum sl 2 at a root of unity. These are generalized quantum invariants depend both on a knot K and a representation of the fundamental group of its complement into SL 2 (C); equivalently, we can think of KR(K) as associating to each knot a function on (a slight generalization of) its character variety. In this paper we clarify some details of their construction. In particular, we show that for K a hyperbolic knot KaRe(K) can be viewed as a function on the geometric component of the A-polynomial curve of K. We compute some examples at a third root of unity. Contents 1. Introduction 1.1. Reshetikhin-Turaev and Kashaev-Reshetikhin invariants 1.2. Geometric interpretation 1.3. Relationship with other work Acknowledgements 2. Quantum sl 2 at a root of unity 2.1. The quantized function algebra 2.2. The Casimir and eigenvalues 3. Representations of the quantum group 4. Braiding on the quantized function algebra 5. Colored tangle diagrams 6. Construction of the invariant 7. The representation variety, the character variety, and the A-polynomial 8. Examples 8.1. The trefoil knot 8.2. The figure eight knot 8.3. Twist knots at the hyperbolic holonomy References