CSQ-ComplexVolume
Reconstructing complex-volume invariants from quandle and 4-fold symmetric quandle cohomology.
Master's Thesis / Early PhD
Suited for students with strong preparation in geometric topology, abstract algebra, and hyperbolic geometry.
12–24 Months
Two individually defensible thesis tracks converging on shared computational infrastructure and a joint paper.
Dr. Alejandro Ucan-Puc
Tecnológico de Monterrey, Campus Monterrey (Department of Mathematics & Data Science).
Executive Summary
For an oriented hyperbolic 3-manifold $M$, the hyperbolic volume $\operatorname{Vol}(M)$ and the Chern–Simons invariant $\operatorname{CS}(M)$ combine into a complex-volume-type invariant. Hyperbolic volume has been interpreted as a quandle cocycle invariant for hyperbolic knots (Inoue & Kabaya 2008, 2014), while constructions on 4-fold symmetric quandles reformulate the Chern–Simons invariant of closed 3-manifolds through quandle cocycles (Hatakenaka & Nosaka 2012, Nosaka 2011).
The central objective of this project is to build an explicit, convention-fixed, and computationally validated bridge between quandle cohomology and complex volume. Specifically, the project targets cusped 3-manifolds and boundary-parabolic $\operatorname{PSL}(2,\mathbb{C})$-representations, comparing quandle pairings directly with Zickert's simplicial formulas and the extended Bloch group.
$$I_Q(M,\rho) \equiv \operatorname{CV}(M,\rho) \pmod{\Lambda}$$
where $\operatorname{CV}(M,\rho)$ denotes a fixed normalization of complex volume (such as Zickert's simplicial regulator) and $\Lambda$ is its period lattice (e.g. $\pi^2\mathbb{Z}$ or $2\pi^2\mathbb{Z}$).Aims and Objectives
Primary Aim
Construct an explicit complex-valued quandle or symmetric quandle cocycle pairing whose evaluation on shadow colorings recovers both hyperbolic volume and the Chern–Simons contribution for hyperbolic knot complements and their Dehn fillings.
Secondary Objectives
- Connect Diagrammatic and Geometric Invariants: Bridge diagrammatic knot invariants, quandle cohomology, ideal triangulations, and secondary characteristic classes of hyperbolic 3-manifolds.
- Computable Realization: Provide an explicit algebraic-topological realization of secondary classes that can be computed algorithmically from knot diagrams and triangulations.
- Link with the Extended Bloch Group: Connect quandle homology directly to the extended Bloch group $\mathcal{B}_{\mathrm{PSL}_2}(\mathbb{C})$ and the Cheeger–Chern–Simons regulator class.
- Extend Beyond Geometric Representations: Investigate boundary-parabolic $\operatorname{PSL}(2,\mathbb{C})$-representations near the geometric component in character varieties.
- Foundations for Higher Structures: Lay groundwork for future research on higher-dimensional hyperbolic volume cocycles, higher-rank representation invariants, and complex Kleinian groups.
Research Questions
| ID | Question | Track Lead |
|---|---|---|
| RQ1 | Which quandle structure best encodes boundary-parabolic $\operatorname{PSL}(2,\mathbb{C})$-representations? | Joint |
| RQ2 | Can one define an explicit complex-valued quandle cocycle? | Tracks A & B |
| RQ3 | How does the quandle cocycle evaluation compare with the extended Bloch-group element and Cheeger–Chern–Simons regulator associated with a representation? | Tracks A & B |
| RQ4 | What are the precise period lattice $\Lambda$, orientation conventions, lifting choices, and branch cuts needed for a well-defined comparison theorem? | Joint (Phase 1) |
| RQ5 | Does the complex quandle invariant distinguish manifolds or representations with equal hyperbolic volume but distinct Chern–Simons data? | Joint |
| RQ6 | How does the invariant behave under Dehn filling, orientation reversal, complex conjugation of representations, and mutation? | Student Track B |
Theoretical Hypotheses
Chern–Simons via Symmetric Quandles
For a suitably chosen 4-fold symmetric quandle and cocycle representative, the quandle cocycle pairing reproduces a fixed normalization of the Chern–Simons invariant of a closed hyperbolic 3-manifold modulo the correct period lattice.
Unified Complex Cocycle
A compatible extension of the Inoue–Kabaya volume cocycle and a symmetric-quandle Chern–Simons cocycle can be packaged into a complex-valued cocycle pairing representing complex volume.
Discriminating Power
The complex-valued quandle invariant contains strictly more topological information than hyperbolic volume alone, separating pairs of representations with identical volumes but distinct Chern–Simons values.
Simplicial Comparison
For boundary-parabolic representations, the quandle construction maps chain-wise to Zickert's decorated ideal triangulation complex and extended Bloch-group regulator under explicit flattening hypotheses.
Computational Benchmark Suite
The theoretical constructions are validated against reference calculations on curated hyperbolic families:
- Figure-Eight Knot Complement ($4_1$): Standard initial testbed with well-documented volume $\operatorname{Vol}(4_1) \approx 2.0298832128$ and vanishing Chern–Simons invariant.
- Twist Knot Complements ($5_2, 6_1, \dots$): Systematic family with accessible Wirtinger presentations and non-trivial Chern–Simons values.
- Two-Bridge Knots and Links: Support explicit $\operatorname{PSL}(2,\mathbb{C})$-character variety descriptions and manageable shadow colorings.
- Dehn Surgeries: Hyperbolic manifolds obtained by $(p,q)$-surgery, bridging cusped and closed computations.
- Equal-Volume / Distinct-CS Pairs: Explicit pairs used to test whether the complex quandle pairing discriminates topological information invisible to volume alone.
Expected Outcomes and Publication Plan
Ladder of Project Outcomes
- Minimum Viable Outcome (Month 8): Verified, reproducible computational implementation of Inoue–Kabaya (2014) and Hatakenaka–Nosaka (2012) algorithms, alongside a complete convention concordance.
- Target Outcome (Month 24): Explicit comparison theorems for both cusped and closed families, pinned-down period lattices, proof of discriminating power, and three submitted papers.
- Stretch Outcome: General comparison theorem extending beyond named knot families without case-by-case re-proof, and analysis of non-geometric boundary-parabolic representations.
- Negative-but-Publishable Outcome: Rigorous obstruction proof demonstrating why ordinary quandle cohomology cannot encode the full Chern–Simons invariant without decorated/extended structures.
Publication Strategy
| Paper | Lead Authors | Target Content | Candidate Venues |
|---|---|---|---|
| Paper 1 | Track A Lead | Explicit quandle-cocycle comparison theorem for complex volume of boundary-parabolic representations (twist-knot family) | J. Knot Theory Ramifications, Topology Appl. |
| Paper 2 | Track B Lead | Explicit 4-fold symmetric quandle Chern–Simons formula for closed manifolds and the Dehn-filling bridge | J. Knot Theory Ramifications, Topology Appl. |
| Paper 3 | Joint | Computational framework, benchmark dataset, high-precision dilogarithm tracking, and distinguishing-power results | Experimental Mathematics, J. Appl. Comput. Topology |
Contact and Inquiries
Interested graduate students (prospective master's or PhD candidates) and researchers interested in collaborating on quandle cohomology, geometric topology, or computational invariants are encouraged to get in touch.
Project Supervision and Contact
Dr. Alejandro Ucan-Puc
Department of Science, Tecnológico de Monterrey (Campus Monterrey)
Office: A7-222
- Email: alejandro.ucan-puc@tec.mx
- ORCID: 0000-0002-0037-9394
- Institutional Profile: research.tec.mx (PID 318207)
- General Contact Page: Contact Alejandro Ucan-Puc
When inquiring about student positions, please mention your background in topology, abstract algebra, and any programming experience in Python/SageMath.